Connectedness of inverse limits with functions fi where

either fi or f−1i is a union of continuum-valued functions

M. M. Marsh

Department of Mathematics & StatisticsCalifornia State University, Sacramento

Sacramento,CA 95819-6051

Abstract

We establish a general theorem for connectedness of the inverse limit Xof an inverse sequence {Xi, fi}i≥1 on metric continua with surjective uppersemi-continuous set-valued bonding functions, where for each i ≥ 1, fi has aconnected graph, and either fi or f−1

i is a union of continuum-valued func-tions. Properties of certain set-valued functions from the factor spaces ontopartial graphs in the inverse sequence imply connectedness of X.

Keywords: connected, inverse limit, partial graph, set-valued function,union of continuum-valued functions2008 MSC: 54C60, 54F15, 54B10, 54D80

1. Introduction

In the setting of inverse sequences {Xi, fi}i≥1 on metric continua withsurjective upper semi-continuous set-valued bonding functions, where, foreach i ≥ 1, fi has a connected graph, and either fi or f−1

i is a union ofcontinuum-valued functions, we prove several theorems for connectedness ofthe partial graphs G′(f1, . . . , fn), and of the inverse limit lim

←−{Xi, fi}.

Properties related to the partial graphs that run from bonding functionsfi that are not unions of continuum-valued functions to bonding functionsfj (i < j), where f−1

j is not a union of continuum-valued functions, will becritical in determining connectedness of the inverse limit. If there are no such

Email address: mmarsh@csus.edu (M. M. Marsh)

Preprint submitted to Topology and Its Applications June 29, 2019

changes in the inverse sequence, as in Corollary 1 or Corollary 3, then theinverse limit is connected.

Our most general results are Theorem 5 and Corollary 4, but other the-orems and corollaries herein may be useful for particular applications. Wegive examples in Section 4 which show that, in general, inverse limits of thesetypes will not be connected. It is known (see Example 2) that the inverselimit may not be connected, even if for each i ≥ 1, either fi or f−1

i is itselfcontinuum-valued. So, additional conditions are necessary. We give suchconditions in Theorem 5, and in Corollaries 4, 5, 6, and 7.

A general introduction to results and questions related to connectednessof an inverse limit with set-valued functions can be found in Section 2 of [7],in Sections 2.2, 2.3, 2.4, 2.6, and 2.7 of [8], and in Section 4 of [10].

Since the inception of “generalized” inverse limits, there has been con-siderable interest in conditions that will ensure connectedness of the inverselimit. Our interest is specialized to those inverse sequences on continua wherefor each i ≥ 1, the graph of the bonding function fi is connected, and eitherfi or its inverse is a union of continuum-valued functions. Although not spe-cific to this area, we would be remiss to not mention the very nice resultson connectedness/nonconnectedness of generalized inverse limits due to SinaGreenwood, Judy Kennedy, and Michael Lockyer, see [1, 2, 3, 4, 5]. Theresults in the first, second, third, and fifth references are for inverse limitson intervals. The results in the fourth reference generalize to inverse limitson continua. Their results include several characterizations of nonconnect-edness of generalized inverse limits. These characterizations are related tothe existence of a certain finite sequence in the infinite product of the fac-tor spaces, called variously C-sequences, CC-sequences, and HC-sequences inthe cited papers. The definitions are somewhat technical, but simply statedthe sequence required for nonconnectedness is related to the existence of aproper basic open set U and a closed set K ⊂ U in the infinite product of thefactor spaces, where the inverse limit meets K and U in the same set, andthe inverse limit is not contained in U . In the fifth reference, componentsof the inverse limit are discussed and a characterization of nonconnectednessis given in terms of the system admitting a component base. See the citedpapers for details and precise defintions.

For inverse limits on [0, 1] with a single upper semi-continuous bondingfunction f , in [6, Theorems 4.2 and 4.3], Ingram established some connected-ness results if f is a union of mappings. In [7], with f a union of two interval-

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valued functions, one of which is surjective, Ingram shows that lim←−{[0, 1], f}

is connected.In the Hausdorff setting, Ingram and Mahavier [10, Theorems 4.7 and

4.8] proved that inverse limits on continua, where all bonding functions arecontinuum-valued or the inverses of all bonding functions are continuum-valued, are connected. Also in the Hausdorff setting, Ingram proved [7,Theorem 2.12] that if X is a continuum and f :X → 2X is a set-valuedfunction with a closed graph that is the union of continuum-valued functions,one of which is surjective and universal with respect to the others, thenlim←−{X, f} is connected.

In [14], Nall proved three theorems related to an inverse limit on a singlecontinuum with a single surjective upper semi-continuous bonding function.His results are also established in the Hausdorff setting. They are

(1) (Theorem 3.1) If X is a continuum and f :X → 2X is a union ofcontinuum-valued functions, then lim

←−{X, f} is connected.

(2) (Theorem 3.3) Suppose X is a continuum and f :X → 2X is surjective.Then lim

←−{X, f} is connected if and only if lim

←−{X, f−1} is connected.

(3) (Theorem 3.5) Suppose X is a continuum, f :X → 2X is surjective,and lim

←−{X, f} is connected. If g:X → X is a mapping that commutes with

f , and the graphs of f and g are not disjoint, then lim←−{X, f∪g} is connected.

In [12], the author generalized Nall’s Theorem 3.1 above, in the metricsetting, to inverse limits on inverse sequences of continua Xi, and bondingfunctions where either all fi or all f−1

i are unions of continuum-valued func-tions. Since the author’s notation in that paper was a bit different than inthis paper, we include, in Section 5, the results related to connectedness inSection 1 of [12]. Also, it will be convenient for the reader to have all resultsthat are used in this paper readily at hand. These results and a few otherearly results about connectedness of both partial graphs and inverse limitsin this setting are also given in Section 5. First, we introduce the relevantdefinitions and notation.

2. Definitions and remarks

A compactum is a compact metric space. All spaces considered in this pa-per will be compacta. A continuum is a connected compactum. A continuousfunction with be referred to as a mapping.

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Let X and Y be compacta. A function f :X → 2Y is upper semi-continuous at the point x ∈ X if for each open set V in Y containing the setf(x), there is an open set U in X such that x ∈ U and f(p) ⊂ V for eachp ∈ U . If f :X → 2Y is upper semi-continuous at each point of X, then fis said to be upper semi-continuous. Ingram shows in [8, Theorem 1.2], thatthe function f :X → 2Y being upper semi-continuous is equivalent to havingthe graph of f be closed. We will occasionally apply this theorem withoutcomment.

We refer to functions f :X → 2Y as set-valued functions from X to Y andwe write f :X → Y is a set-valued function. The graph of f , which we denoteby G(f), is the subset of X × Y consisting of all points (x, y) with y ∈ f(x).For each product X × Y of compacta X and Y , let c2:X × Y → Y denotecoordinate projection. The range of the set-valued function f :X → Y isdefined as R(f) = c2(G(f)). The set-valued function f :X → Y is surjectiveif R(f) = Y . If A ⊂ X, let f |A be the set-valued function whose domain isA and such that f |A(x) = f(x) for x ∈ A.

A set-valued function f :X → Y is continuum-valued if for each x ∈ X,the set f(x) is a subcontinuum of Y . Suppose that f :X → Y is a set-valuedfunction, and g′:X → Y is a continuum-valued function with G(g′) ⊂ G(f).We say that g′:X → Y is max continuum-valued if whenever g:X → Y iscontinuum-valued with G(g′) ⊂ G(g) ⊂ G(f), we have that G(g) = G(g′).We note that the graph of each continuum-valued function g such thatG(g) ⊂G(f) is contained in the graph of a max continuum-valued function g′ suchthat G(g′) ⊂ G(f). Specifically, for each x ∈ X, simply define g′(x) to bec2(L), where L is the component of ({x}×Y )∩G(f) that contains {x}×g(x).If there exists x ∈ X such that f(x) = Y , we say that f is full-valued at x.

We say that a set-valued function f :X → Y is a union of continuum-valued functions if for each x ∈ X and each y ∈ f(x), there exists acontinuum-valued function g:X → Y such that y ∈ g(x) and G(g) ⊂ G(f).If the set-valued function f :X → Y is surjective, we say that f−1:Y → Xis a union of continuum-valued functions if for each y ∈ Y and x ∈ f−1(y),there exists a continuum-valued function g:Y → X such that x ∈ g(y) andG(g) ⊂ G(f−1).

Suppose f :X → Y is a union of continuum-valued functions. Let C(f)be the collection of all max continuum-valued functions g:X → Y such thatG(g) ⊂ G(f). If there exists h ∈ C(f) such that whenever g ∈ C(f), we havethat G(g) ∩ G(h) 6= ∅, then we say that h is universal with respect to eachg ∈ C(f). Under these conditions, we say that C(f) has a universal member.

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A minor variation of this definition was introduced by Ingram in [7]. We coulddefine C(f) to be the collection of all continuum-valued functions g:X → Ysuch that G(g) ⊂ G(f). The existence, or non-existence, of a universalmember of C(f) would not be changed. We prefer the given definition asit is typically easier, for specific examples, to identify the collection of maxcontinuum-valued functions. We note that if C(f) has a universal member,then G(f) is connected.

Let X1, X2, . . . be a sequence of compacta. Throughout, we let {Xi, fi}i≥1

denote an inverse sequence with upper semi-continuous set-valued bondingfunctions fi:Xi+1 → Xi, and its inverse limit is given by

lim←−{Xi, fi} = {x = (x1, x2, . . .) ∈

∏i≥1

Xi | xi ∈ fi(xi+1) for i ≥ 1}.

For j, n ∈ N with j ≤ n, we define the set below.

Gn+1j = G′(fj, . . . , fn) = {x ∈

n+1∏i=j

Xi | xi ∈ fi(xi+1) for j ≤ i ≤ n}.

We refer to these sets as partial graphs in the inverse sequence. Thesesets have also been called Mahavier products and Ingram-Mahavier products.We note that G2

1 = G′(f1) = G(f−11 ). For consistency of notation, for i ≥ 1,

we let Gii = Xi. We emphasize that hereafter all set-valued functions are

assumed to be upper semi-continuous, and to have values that are closed

sets. The notation XT≈ Y will indicate that X is homeomorphic to Y .

Suppose that {Xi}i≥1 is a sequence of compacta, and for each i ≥ 1,fi:Xi+1 → Xi is a surjective, set-valued function. We make the followingdefinitions of functions that will be useful throughout the paper. Fix n ≥ 1.For 1 ≤ i ≤ j ≤ n+ 1, we let

(i) πj be the projection mapping from∏n+1

k=i Xk onto the jth coordinate,

(ii) π(i) be the projection mapping from∏n+1

k=1 Xk onto all coordinates ex-cept the ith coordinate,

(iii) Fi,j:Xj+1 → Gji be the function where (xi, . . . , xj) is in Fi,j(xj+1) if and

only if (xi, xi+1, . . . , xj+1) ∈ Gj+1i , and

(iv) Lj:Gn+1j+1 → Xj be the function where xj is in Lj(xj+1, . . . , xn+1) if and

only if (xj, xj+1, . . . , xn+1) ∈ Gn+1j

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Note that we must specify some n ≥ 1 before the domains of the functionsin (i), (ii), and (iv) above are clear. We use this convention throughout ratherthan making the notation more cumbersome.

Remarks.

1. The set-valued function F1,j:Xj+1 → Gj1 defined in (iii) above is the

function Fj considered by Nall in [13]. Nall proves in Lemma 5.1 thatFj is upper semi-continuous. We will also need the function Lj, definedin (iv) above, in the proofs of Lemma 5 and Theorem 4. That Fi,j andLj are upper semi-continuous follows analogously as in Nall’s proof.

2. In Corollary 4.2 of [9], Ingram establishes that the inverse limit X of aninverse sequence {Xi, fi}i≥1 with set-valued functions is homeomorphicto an inverse limit on the sequence of partial graphs {Gi

1}i≥1 withbonding functions that are surjective mappings. We will use this resultin the proof of several corollaries. Also, we observe that since Gi

1 isa continuous image of Gi+1

1 for each i ≥ 1, it follows that if Gn1 is

connected for some n ≥ 1, then Gi1 is connected for each 1 ≤ i ≤ n.

The next remark shows that certain amalgamations of portions of a finiteinverse sequence produce a shorter inverse sequence whose partial graph ishomeomorphic to the partial graph of the original sequence.

3. Suppose {Xi, fi}ni=1 is a finite inverse sequence on compacta, where foreach 1 ≤ i ≤ n, fi is a surjective set-valued function. Let 1 ≤ j < k ≤n. Then Gn+1

1 is homeomorphic to the partial graph of each inversesequence below.

(1) Gj1

F1,j←− Xj+1fj+1←− . . .

fn←− Xn+1, and

(2) X1f1←− . . .

fj−1←− Xj

πj |Gkj←− Gk

j

Fj,k←− Xk+1fk+1←− . . .

fn←− Xn+1.

It is clear that Gn+11 is homeomorphic to the partial graph in (1). Let

G denote the partial graph of the sequence in (2). That Gn+11

T≈ G

is easily seen by noting that the point (x1, . . . , xj, xj, . . . , xk, . . . , xn+1)is in G if and only if the point (x1, . . . , xj, xj+1, . . . , xk, . . . , xn+1) isin Gn+1

1 , and that the mapping h:G → Gn+11 that drops one of the

repeated jth-coordinates is a homeomorphism.

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Our objective is to find reasons why some inverse limits on continua areconnected, and others are not, when the bonding functions or their inversesare unions of continuum-valued functions. In this setting, we introduce ter-minology that should provide immediate clarity and replace the use of long,cumbersome statements. The terminology will also benefit discussion of theexamples in Section 4. Although the definitions below are for infinite inversesequences, we will use analogous terminology for finite inverse sequences.

Let {Xi, fi}i≥1 be an inverse sequence, where for each i ≥ 1, Xi is acontinuum, fi:Xi+1 → Xi is a surjective set-valued function with a connectedgraph, and either fi or f−1

i is a union of continuum-valued functions.For a given i ≥ 1, if fi is a union of continuum-valued functions, we say

that fi is a union; if f−1i is a union of continuum-valued functions, we say

that fi is an inverse union. We note that under our general assumption, iffi is not a union, then fi is an inverse union, and vice versa.

If, in the inverse sequence {Xi, fi}i≥1, there exist i and j where fi is not aunion and fj is not an inverse union, then we call {Xi, fi}i≥1 a mixed inversesequence. If i < j, we say that there is a change from an inverse union (at i)to a union (at j); and if j < i, we say that there is a change from a union (atj) to an inverse union (at i). As we will see, a change from a union (at j) toan inverse union (at i), that has no other changes from j to i, will have itsassociated partial graph Gi+1

j be connected. This, in general, is not the casefor changes from an inverse union to a union. So, additional conditions mustbe assumed to ensure connectedness of partial graphs of inverse sequencesthat contain such changes. We provide one such condition, but, of course,there may be others.

It would be of interest to know if any condition related to the compositionfunction fi,j = fi ◦ fi+1 ◦ · · · ◦ fj−1 when i < j− 1, through a change from aninverse union (at i) to a union (at j) could ensure connectedness of Gj+1

i . Inthe first submission of this paper, the author claimed to have such a condition,but a careful and thorough referee observed that a critical theorem relatedto the composition property was incorrect. It seems that any condition onthe composition function fi,j that is related to our general assumption ofbeing either a union or an inverse union of continuum-valued functions is notsufficient. In fact, even if j = i+ 2 and the composition function fi,j is botha union and an inverse union, the partial graph Gj

i may not be connected.Too much structural information about the partial graph Gj

i is lost in thegraph of fi,j. The author is indepted to the referee for finding this error.

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3. Determining sequences

Let {Xi, fi}i≥1 be a mixed inverse sequence containing a change from aninverse union (at j) to a union (at k). We say that the sequence fj, . . . , fkis a determining sequence if either j = 1 or fj−1 is a union, and fi is aninverse union for all j ≤ i < k. A max determining sequence is a determin-ing sequence that is not a “proper” subsequence of any other determiningsequence.

A mixed inverse sequence {Xi, fi}i≥1 is eventually alternating if thereexists k ≥ 1 such that for all i ≥ 0, fk+2i is not a union, and fk+2i+1 is not aninverse union. If k = 1, we say that {Xi, fi}i≥1 is alternating. We note thatin an eventually alternating inverse sequence each successive pair of bondingfunctions starting with fk+2 and fk+3 is a max determining sequence. Ifk = 1, then f1, f2 is also a max determining sequence.

Since conditions on determining sequences will be a central part of mostof our main results, we make some observations related to determining se-quences.

Observation 1. If {Xi, fi}i≥1 is a mixed inverse sequence containing achange from an inverse union (at `) to a union (at m), then there existsa determining sequence in f1, . . . , fm.

Proof. Let k be the least integer in {` + 1, . . . ,m} such that fk is not aninverse union. Since fm is not an inverse union, k must exist. So, we havethat for ` ≤ i < k, fi is an inverse union. If ` = 1, or if, for all 1 ≤ i < `,fi is not a union, then by definition, f1, . . . , fk is a determining sequence.Otherwise, pick the largest j, where 1 < j ≤ ` and fj−1 is a union. Thenfj, . . . , fk is a determining sequence.

Observation 2. If {Xi, fi}i≥1 is a mixed inverse sequence containing twodetermining sequences fj, . . . , fk and f`, . . . , fm with j ≤ `, then either k < `or k = m.

Proof. Note that k 6= ` since fk is not an inverse union and f` is an inverseunion. So, we assume that ` < k.

If j = `, assume, without loss of generality, that k ≤ m. If k = m,the proof is complete. If k < m, then the determining sequence f`, . . . , fmcontains fk, which is not an inverse union, contradicting that f`, . . . , fm is adetermining sequence.

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So, we have that j < ` < k. If either m < k or k < m, we violate thedefinition of determining sequence, as in the previous paragraph, for one ofthe two sequences. So, k = m and the proof is complete.

It follows from Observation 2 that each determining sequence is containedin a max determining sequence. Also, if fj, . . . , fk and f`, . . . , fm are maxdetermining sequences with j < `, then k < `. We say, in this case, thatthe two max determining sequences are disjoint, and we note that each twodistinct max determining sequences are disjoint. If there is no determiningsequence in fk+1, . . . , f`−1, we say that the two max determining sequencesare consecutive. We say that fj, . . . , fk and f`, . . . , fm are adjacent if ` = k+1.

Observation 3. Suppose that {Xi, fi}ni=1 is a finite mixed inverse sequenceon continua. Suppose also that j > 1, and fj, . . . , fk is the first max de-termining sequence in f1, . . . , fn, in the sense that there is no determiningsequence in f1, . . . , fj−1. Then for all 1 ≤ i ≤ j − 1, fi is a union.

Proof. Suppose there exists m with 1 ≤ m ≤ j − 1 and fm is not a union.We assume that m is the least such integer. So, either m = 1, or for each1 ≤ i < m, fi is a union. Since fj, . . . , fk is a max determining sequence,it follows that there exists m < r ≤ j − 1 where fr is not an inverse union;for otherwise, fm, . . . , fk is a determining sequence “properly” containingfj, . . . , fk. But now we have that fm, . . . , fr is a determining sequence inf1, . . . , fj−1, which contradicts our hypothesis.

Observation 4. Suppose that {Xi, fi}ni=1 is a finite mixed inverse sequenceon continua containing two consecutive max determining sequences fj, . . . , fkand f`, . . . , fm with k < `. Then fi is a union for all k + 1 ≤ i ≤ `− 1.

Proof. The proof is analogous to the proof of Observation 3.

Observation 5. Suppose {Xi, fi}i≥1 is an inverse sequence on continua,where for each i ≥ 1, fi is either a union or an inverse union. Suppose alsothat {Xi, fi}i≥1 contains no determining sequence. Then either

(i) fi is an inverse union for all i ≥ 1,

(ii) fi is a union for all i ≥ 1, or

(iii) there exists k > 1 such that fi is a union for all 1 ≤ i ≤ k − 1, and fiis an inverse union for all i ≥ k.

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Proof. Suppose that (i) and (ii) are not the case. Since (i) is not the case, forsome j ≥ 1, fj is not an inverse union. So, fj is a union, and for 1 ≤ i < j,fi is a union. For otherwise, by Observation 1 there would be a determiningsequence in f1, . . . , fj. Since (ii) is not the case, for some k > j, fk is not aunion. We assume that k is the least such integer. So, we have that for all1 ≤ i < k, fi is a union. Suppose there exists m > k such that fm is not aninverse union. Then, again by Observation 1, there would be a determiningsequence in f1, . . . , fm, contradicting the hypothesis. Hence, we have that(iii) holds.

Observation 6. Suppose {Xi, fi}i≥1 is an inverse sequence on continua,where for each i ≥ 1, fi is surjective with a connected graph, and fi iseither a union or an inverse union. Suppose also that {Xi, fi}i≥1 containsno determining sequence. Then lim

←−{Xi, fi} is a continuum.

Proof. See Corollaries 1 and 3 in Sections 4 and 5 respectively.

4. Examples

As we will see in the examples below, the determining sequences maydestroy the connectedness of a partial graph, and in turn, of the inverselimit. The additional condition in Theorem 5 will ensure that the partialgraphs will remain connected through max determining sequences, and thecondition will also ensure connectedness of the partial graphs Gn

1 and of theinverse limit. So, it is the determining sequences that will, indeed, determineconnectedness of inverse limits of mixed inverse sequences if certain propertiesare satisfied. Results in Section 6 will provide sufficient conditions on maxdetermining sequences to establish connectedness of the partial graphs Gn

1

and of the inverse limit space.For readers familiar with techniques introduced in [4] and [5], connect-

edness (or non-connectedness) of the partial graphs associated with the maxdetermining sequences could also be determined by applying methods from[4] in the case of inverse sequences on continua, and from [5] in the caseof inverse sequences on intervals. However, Example 3 illustrates that con-nectedness alone of the partial graphs associated with the max determiningsequences in f1, f2, . . . , fn is not enough to determine connectedness of Gn

1 .So, the additional contition in Theorem 5 must be checked for the max deter-mining sequences to establish connectedness of the inverse limit space. We

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discuss these ideas in the examples below. In each connected example in thissection, we reference results in later sections that establish connectedness ofthe example.

Example 1. (Example 2.3 in [11]) In this example, set-valued functionsf1, f2: [0, 1]→ [0, 1] are defined by Ingram and Marsh (see Figure 1) in sucha way that f1 is a union of two mappings (and not an inverse union), and f2

is an inverse union of two mappings (and not a union). For i ≥ 3, let fi bethe identity mapping on [0, 1].

��������

������

��

��������

��������

f1 f2

Figure 1. Graphs of the functions in Example 1

Let X1, X2, and X3 be, respectively, the limits of the following threeinverse sequences,

[0, 1]f1←− [0, 1]

f2←− [0, 1]f3←− . . . . . . , [0, 1]

f2←− [0, 1]f1←− [0, 1]

f3←− . . . . . . ,

and [0, 1]f−11←− [0, 1]

f−12←− [0, 1]

f3←− . . . . . .

Ingram and Marsh show that X1 is connected, while X2 and X3 are notconnected. We note that, in the first inverse sequence, there is a change fromf1 a union to f2 an inverse union, and there are no other changes. This typeof change preserves the connectedness of the partial graph G3

1 (see Corollary

3, Section 6). For n ≥ 3, Gn1

T≈ G3

1, so X1 is connected.In the second and third inverse sequences, there is a change from an

inverse union to a union, respectively f2, f1 and f−11 , f−1

2 , and there are noother changes. The partial graph G3

1 in each of these two inverse sequences is

not connected. Again, for n ≥ 3, Gn1

T≈ G3

1, so X2 and X3 are not connected.

Even when the bonding functions, or their inverses, are continuum-valued,a determining sequence can destroy connectedness of the associated partial

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graph. The next example illustrates this, but we indicate how an additionalcondition added to the determining sequence can ensure connectedness of thepartial graph. For this example, the condition will be that F1,2: [0, 1]→ G2

1 isa union, and C(F1,2) has a universal member (see Corollaries 6 and 7, Section6).

Example 2. Let f1: [0, 1] → [0, 1] be the set-valued function defined byf1(t) = {1

2} for 0 ≤ t < 3

4, and f1(t) = {1

2, 4t − 3} for 3

4≤ t ≤ 1. Note

that f−11 is given by f−1

1 (12) = [0, 1], and otherwise f−1

1 (t) = { t+34}. So, f1 is

not a union, but f−11 is, in fact, continuum-valued. Let f2: [0, 1] → [0, 1] be

defined by f2(12) = [0, 1], and otherwise, f2(t) = {1

2t + 1

4}. We see that f2 is

continuum-valued, and f−12 is not a union (see Figure 2). For i ≥ 3, let fi be

the identity mapping on [0, 1].

��������

�����

���

�����

���

f1 f2 g2

Figure 2. Graphs of the functions in Example 2

As in the second and third inverse sequences from Example 1, we have amax determining sequence f1, f2, and, indeed, the partial graph G3

1 and theinverse limit are not connected. If we add identity mappings between f1 andf2 , for example, as in the inverse sequence

[0, 1]f1←− [0, 1]

id←− [0, 1]id←− [0, 1]

f2←− [0, 1]f3←− . . . . . . ,

the change from inverse union to union has been “spread out”. Now, themax determining sequence runs from the first bonding function through thefourth. All 3-length partial graphs are connected, but the partial graph G5

1,which contains the change, is not connected. So, if we wish to add conditionsto ensure connectedness of the inverse limit, we must account for changes ofthis type that happen “far apart” in the inverse sequence.

In the original sequence, if we replace f2 with g2 defined below, we stillhave a max determining sequence from 1 to 2, but G′(f1, g2) will be con-nected.

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Define f ′2: [0, 1]→ [0, 1] by f ′2(t) = {12t + 1

4} for 0 ≤ t < 1

2, f ′2(1

2) = [1

2, 3

4],

and f ′2(t) = {34} for 1

2< t ≤ 1. Note that f ′2 is continuum-valued. Let

g2 be the set-valued function whose graph is G(f2) ∪ G(f ′2) (see Figure 2).By definition, g2 is a union of continuum-valued functions. We note that

F1,2: [0, 1]→ G21, for the finite inverse sequence [0, 1]

f1←− [0, 1]g2←− [0, 1], is a

union, and C(F1,2) has a universal member.Specifically, F1,2 is a union of the three max continuum-valued functions

h1, h2, and h3 defined as follows.Let h1(t) = {(1

2, 1

2t + 1

4)} for 0 ≤ t < 1

2or 1

2< t ≤ 1, and let h1(1

2) =

G21 = G(f−1

1 ).Let h2(t) = {(1

2, 1

2t+ 1

4)} for 0 ≤ t < 1

2, h2(1

2) = G2

1, and h2(t) = {(12, 3

4)}

for 12< t ≤ 1.

Let h3(t) = {(12, 1

2t+ 1

4)} for 0 ≤ t < 1

2, h3(1

2) = G2

1, and h3(t) = {(0, 34)}

for 12< t ≤ 1.

Clearly, each pair in {G(h1), G(h2), G(h3)} has the set {12} ×G2

1 in com-mon. So, each hi is universal with respect to the other two. It follows that

G′(f1, g2) is connected, and since Gn1

T≈ G3

1 for n ≥ 3, the limit of the inversesequence

[0, 1]f1←− [0, 1]

g2←− [0, 1]f3←− [0, 1]

f4←− . . . . . .

is connected (See Corollaries 6 and 7, Section 6).A somewhat simpler revision can be made to f2 (see f̂2 defined below) so

that F1,2 will itself be continuum-valued. Hence, G′(f1, f̂2)T≈ G(F1,2) will be

connected by Lemma 1. Let f̂2(t) = f2(t) for 0 ≤ t < 1, and let f̂2(1) = [34, 1].

One might ask if only having the partial graphs Gk+1j connected for

each max determining sequence fj, . . . , fk in a finite mixed inverse sequence{Xi, fi}ni=1 is enough to ensure connectedness of Gn+1

1 . We show in Example3 that this is not the case.

Example 3. Let k1: [0, 1] → [0, 1] be the set-valued function defined byk1(t) = {1

2} for 0 ≤ t < 3

4, k1(t) = {1

2, 4t − 3} for 3

4≤ t ≤ 7

8, and k1(t) =

{4t − 3} for 78< t ≤ 1. We see that k1 is not a union, but k−1

1 is, infact, continuum-valued. Let k2: [0, 1] → [0, 1] be defined by k2(t) = {t} for0 ≤ t < 7

8, k2(7

8) = [7

8, 1], and k2(t) = {7

4− t} for 7

8< t ≤ 1. We see that

f2 is continuum-valued, and f2 is not an inverse union. Let k3: [0, 1]→ [0, 1]be defined by k3(t) = {t} for 0 ≤ t < 1

4or 1

2< t ≤ 1, and k3(t) = {t, 1

2} for

14≤ t ≤ 1

2. Finally, let k4 be the continuum-valued function f2 defined in

Example 2.

13

��������

�������@

��������

k1 k2 k3

Figure 3. Graphs of the functions in Example 3

Suppose we have the finite alternating inverse sequence below.

[0, 1]k1←− [0, 1]

k2←− [0, 1]k3←− [0, 1]

k4←− [0, 1]

It is straightforward to check that k1, k2 and k3, k4 are adjacent max deter-mining sequences, and that G3

1 and G53 are connected. We leave to the reader

to check that (0, 34, 3

4, 3

4, 1) is an isolated point in G5

1; or, using techniquesfrom [5], one may note that {〈3

4, 0〉, 〈3

4, 3

4〉, 〈3

4, 3

4〉, 〈3

4, 1〉} is a component base

for the functions k1, k2, k3, and k4. So, G51 is not connected. It is also inter-

esting to note that F3,4: [0, 1] → G43 is a union, and C(F3,4) has a universal

member, but F1,2: [0, 1]→ G21 is not a union.

5. Results related to our setting

Lemma 1 below is Theorem 4.1 in [10].

Lemma 1. If f :X2 → X1 is a continuum-valued function and X2 is con-nected, then G(f) is connected.

The next Lemma is clear.

Lemma 2. If the set-valued function f :X2 → X1 is a union of continuum-valued functions, then for each closed subset K of X2, f |K :K → X1 is aunion of continuum-valued functions.

Lemma 3 below follows from Theorems 4.3 and 4.5 in [10].

Lemma 3. Suppose X1, X2, . . . , Xk+1 are continua and for each 1 ≤ i ≤ k,fi:Xi+1 → Xi is a surjective set-valued function. If each fi is continuum-valued or if each f−1

i is continuum-valued, then Gk+11 is connected.

14

Lemma 4. Suppose that X1, X2, . . . , Xn+1 are continua and for each 1 ≤ i ≤n, fi:Xi+1 → Xi is a surjective set-valued function whose graph is connected.Suppose also that for each 1 ≤ i ≤ n, fi:Xi+1 → Xi is a union of continuum-valued functions. Then the set-valued function F1,n:Xn+1 → Gn

1 is a unionof continuum-valued functions.

Proof. We use induction on the number of bonding functions in a sequencewhere each bonding function is a union of continuum-valued functions. Ifn = 1, then F1,1 = f1 and f1:X2 → X1 is a union of continuum-valuedfunctions by assumption.

Assume that F1,n−1:Xn → Gn−11 is a union of continuum-valued functions

for some n ≥ 2.Let (z1, z2, . . . , zn) ∈ F1,n(zn+1). Since fn is a union of continuum-valued

functions, there exists a continuum-valued function h:Xn+1 → Xn suchthat zn ∈ h(zn+1) and G(h) ⊂ G(fn). Also, by inductive assumption andLemma 2, there exists a continuum-valued function g:R(h) → Gn−1

1 suchthat (z1, . . . , zn−1) ∈ g(zn) and G(g) ⊂ G(F1,n−1). Since R(h) = c2(G(h)), itfollows that R(h) is closed and connected in Xn.

Consider the set-valued function `:Xn+1 → Gn1 defined by `(x) = G(g|−1

h(x)).

Since both g and h are continuum-valued, it follows by Lemma 1 thatG(g|h(x))is connected. So, G(g|−1

h(x)) is connected. Hence, ` is continuum-valued. Also,

(z1, z2, . . . , zn) ∈ `(zn+1), and G(`) ⊂ G(F1,n). We have that F1,n:Xn+1 →Gn

1 is a union of continuum-valued functions.

Lemma 5. Suppose that X1, X2, . . . , Xn+1 are continua and for each 1 ≤i ≤ n, fi:Xi+1 → Xi is a surjective set-valued function whose graph is con-nected. Suppose also that for each 1 ≤ i ≤ n, f−1

i :Xi → Xi+1 is a union ofcontinuum-valued functions. Then the set-valued function L−1

1 :X1 → Gn+12

is a union of continuum-valued functions.

Proof. We use induction on the number of bonding functions in a sequencewhere the inverse of each bonding function is a union of continuum-valuedfunctions. If n = 1, then we have L−1

1 :X1 → X2, and by definition, L−11 =

f−11 , which is a union of continuum-valued functions by assumption.

For some n ≥ 2, in the sequence X2, . . . , Xn+1 which has n − 1 bond-ing functions, assume that L−1

2 :X2 → Gn+13 is a union of continuum-valued

functions.Let (z2, . . . , zn+1) ∈ L−1

1 (z1). Since f−11 is a union of continuum-valued

functions, there exists a continuum-valued function h:X1 → X2 such that

15

z2 ∈ h(z1) and G(h) ⊂ G(f−11 ). Also, by inductive assumption and Lemma 2,

there is a continuum-valued function g:R(h)→ Gn+13 such that (z3, . . . , zn+1)

is in g(z2) and G(g) ⊂ G(L−12 ). As in the proof of Lemma 4, it follows that

R(h) is closed and connected in X2.Consider the set-valued function `:X1 → Gn+1

2 defined by `(x) = G(g|h(x)).Since both g and h are continuum-valued, it follows by Lemma 1 thatG(g|h(x))is connected. Analogously, as in the proof of Lemma 4, we get that ` iscontinuum-valued, (z2, . . . , zn+1) ∈ `(z1), and G(`) ⊂ G(L−1

1 ). Hence, L−11 is

a union of continuum-valued functions.

Theorem 1. Suppose that X1, X2, . . . , Xn+1 are continua and for each 1 ≤i ≤ n, fi:Xi+1 → Xi is a surjective set-valued function whose graph is con-nected. Suppose also that for each 1 ≤ i ≤ n, fi:Xi+1 → Xi is a union ofcontinuum-valued functions. Then Gn+1

1 is connected.

Proof. We use induction on the number of bonding functions. For n = 1,G2

1 = G(f−11 ) is connected by assumption.

Assume that Gn1 = G′(f1, . . . , fn−1) is connected. Let A and B be

closed sets whose union is Gn+11 . Let ρ:Gn+1

1 → Gn1 be the mapping given

by ρ = π(n+1)|Gn+11

. Since ρ(A) ∪ ρ(B) = Gn1 , there exists a point p =

(p1, . . . , pn) ∈ ρ(A) ∩ ρ(B). Assume a = (p1, . . . , pn, an+1) ∈ A and b =(p1, . . . , pn, bn+1) ∈ B. By inductive assumption, Gn

1 is connected. Also, byLemma 4, F1,n−1:Xn → Gn−1

1 is a union of continuum-valued functions. Letg:Xn → Gn−1

1 be a continuum-valued function such that (p1, . . . , pn−1) ∈g(pn) and G(g) ⊂ G(F1,n−1).

Let `:Gn+1n → Gn−1

1 be defined by `(xn, xn+1) = g(xn). Note that Gn+1n

is connected by assumption, and ` is continuum-valued since g is continuum-valued. It follows from Lemma 1 that G(`) is connected. So, G(`−1) isconnected. Also, we see that a and b are in G(`−1) ⊂ Gn+1

1 . So, G(`−1)meets both A and B. It follows that A and B are not mutually separated.Hence, Gn+1

1 is connected.

Theorem 2. Suppose that X1, X2, . . . , Xn+1 are continua and for each 1 ≤i ≤ n, fi:Xi+1 → Xi is a surjective set-valued function whose graph is con-nected. Suppose also that for each 1 ≤ i ≤ n, f−1

i :Xi → Xi+1 is a union ofcontinuum-valued functions. Then Gn+1

1 is connected.

Proof. The proof of this theorem is similar to the proof of Theorem 1, usingLemma 5 analogously as Lemma 4 was used in Theorem 1.

16

In Theorem 3.1 of [14], Nall has a condition on a surjective relationF :X → X that ensures lim

←−{X,F} is a continuum. His condition that F

be the union of a collection of closed subsets {Fα}α∈Γ with certain propertiesis equivalent to F being a union of continuum-valued functions. Via Nall’sTheorems 3.1 and 3.3 (see items (1) and (2) in the next to last paragraphof the introduction), having F−1 be a union of continuum-valued functionsalso leads to the connectedness of lim

←−{X,F}. We are able to generalize his

Theorem 3.1(see Corollary 1 below) by additionally showing that for dif-ferent factor spaces and different set-valued bonding functions, the inverselimit space lim

←−{Xi, fi} will be a continuum. The inverse limits X1 and X3

in Example 1 show that Nall’s Theorem 3.3 cannot be generalized to eachsequence of set-valued functions.

Corollary 1. Let X1, X2, . . . be a sequence of continua and suppose that foreach i ≥ 1, fi:Xi+1 → Xi is a surjective set-valued function whose graph isconnected. Suppose also that for each i ≥ 1, fi:Xi+1 → Xi is a union ofcontinuum-valued functions, or for each i ≥ 1, f−1

i :Xi → Xi+1 is a union ofcontinuum-valued functions. Then lim

←−{Xi, fi} is a continuum.

Proof. The corollary follows from Theorems 1 and 2, and from Ingram’sresult noted in Remark 2.

6. Main theorems

Corollaries 4 and 7 of this section are our most general results for connect-edness of an inverse limit in our setting, but other theorems and corollariesin this section provide conditions for connectedness of both partial graphsand inverse limits that could be useful in specific cases.

The proofs of Theorems 3 and 4 are similar to the proofs of Lemmas 4 and5, and Theorems 1 and 2 in Section 5. This typically signals some generaltheorem that includes them all as special cases. The author was unable tofind any such easily stated theorem. Perhaps the repetitiveness of the proofswill be helpful rather than bothersome.

Theorem 3 establishes connectedness of the partial graph Gn+1i , when

f−1n is a union of continuum-valued functions, and a partial graph to the left,

namely Gni , is connected. Theorem 4 establishes connectedness of the partial

graph Gn+1k , when fk is a union of continuum-valued functions, and a partial

graph to the right, namely Gn+1k+1 , is connected.

17

Theorem 3. Suppose that X1, X2, . . . , Xn+1 are continua, and for each1 ≤ i ≤ n, fi:Xi+1 → Xi is a surjective set-valued function whose graph isconnected. Suppose also that f−1

n :Xn → Xn+1 is a union of continuum-valuedfunctions. Then, for 1 ≤ i < n,

(1) F−1i,n :Gn

i → Xn+1 is a union of continuum-valued functions,

(2) if Gni is connected, then G(F−1

i,n )T≈ Gn+1

i is connected, and

(3) if Fi,n is a union of continuum-valued functions, then fn is a union ofcontinuum-valued functions.

Proof. (1) Let zn+1 ∈ F−1i,n (zi, . . . , zn). Since zn+1 ∈ f−1

n (zn) and f−1n is a

union of continuum-valued functions, there exists a continuum-valued func-tion g:Xn → Xn+1 such that zn+1 ∈ g(zn) and G(g) ⊂ G(f−1

n ).Let `:Gn

i → Xn+1 be the set-valued function defined by `(xi, . . . , xn) =g(xn). We see that zn+1 ∈ `(zi, . . . , zn), and G(`) ⊂ G(F−1

i,n ). Since g is

continuum-valued, it follows that ` is continuum-valued. So, F−1i,n :Gn

i →Xn+1 is a union of continuum-valued functions.

(2) Let A and B be closed sets such that A∪B = Gn+1i . Let ρ:Gn+1

i → Gn+1n

be projection onto the nth and (n+1)th coordinates. Since G(fn) is connectedand ρ(A)∪ρ(B) = Gn+1

n = G(f−1n ), there is a point (pn, pn+1) ∈ ρ(A)∩ρ(B).

Let (ai, . . . , an−1, pn, pn+1) ∈ A, and (bi, . . . , bn−1, pn, pn+1) ∈ B. We definethe continuum-valued functions g:Xn → Xn+1 and `:Gn

i → Xn+1 with pn+1 ∈g(pn) analogously as in (1). Recall that G(`) ⊂ G(F−1

i,n ) = Gn+1i . Since Gn

i

is connected by hypothesis, it follows from Lemma 1 that G(`) is connected.Also, G(`) intersects both A and B; so, A and B are not mutually separated.Hence, Gn+1

i is connected.

(3) Let zn ∈ fn(zn+1). Let (zi, . . . , zn) ∈ Gni . Then (zi, . . . , zn) ∈ Fi,n(zn+1),

and, by assumption, there exists a continuum-valued g:Xn+1 → Gni such that

(zi, . . . , zn) ∈ g(zn+1) and G(g) ⊂ G(Fi,n). Let g′:Xn+1 → Xn be defined byg′ = πn◦g. We note that g′ is continuum-valued, since g is continuum-valuedand πn is a mapping. Also, zn ∈ g′(zn+1) and G(g′) ⊂ G(fn). So, fn is aunion of continuum-valued functions.

Corollary 2. Suppose that {Xi, fi}i≥1 is an inverse sequence on continua,and for each i ≥ 1, fi:Xi+1 → Xi is a surjective set-valued function whosegraph is connected. Suppose also that there exists k ≥ 1 where Gk

1 is con-nected, and for all i ≥ k, f−1

i :Xi → Xi+1 is a union of continuum-valued

18

functions. Then for each n > k, Gn1 is connected, and hence, lim

←−{Xi, fi} is

a continuum.

Proof. Fix n > k. By Theorem 3, F1,k:Xk+1 → Gk1 is an inverse union with

a connected graph. So, we have that

Gk1

F1,k←− Xk+1fk+1←− . . .

fn←− Xn+1

is a finite inverse sequence, where each bonding function is an inverse unionwith a connected graph. By Theorem 2 and Remark 3, Gn

1 is connected.By Remark 2 and the fact that ordinary inverse limits on continua are

continua, it follows that lim←−{Xi, fi} is a continuum.

Corollary 3. Suppose that {Xi, fi}i≥1 is an inverse sequence on continua,and for each i ≥ 1, fi:Xi+1 → Xi is a surjective set-valued function whosegraph is connected. Suppose also that there exists k ∈ N with k ≥ 2 such that

for each 1 ≤ i < k, fi:Xi+1 → Xi is a union of continuum-valued func-tions, and

for each i ≥ k, f−1i :Xi → Xi+1 is a union of continuum-valued functions.

Then, for each n > k, Gn1 is connected, and hence, lim

←−{Xi, fi} is a contin-

uum.

Proof. By Theorem 1, Gk1 is connected. By Corollary 2, the result follows.

Theorem 4 below is not needed to prove any of the theorems that followit, but we include it and its proof since it could be useful in establishingconnectness of some partial graphs in certain applications. Also, it is a nicecompanion theorem to Theorem 3.

Theorem 4. Suppose that Xk, Xk+1, . . . , Xn+1 are continua, and for eachk ≤ i ≤ n, fi:Xi+1 → Xi is a surjective set-valued function whose graph isconnected. Suppose also that fk:Xk+1 → Xk is a union of continuum-valuedfunctions. Then

(1) Lk:Gn+1k+1 → Xk is a union of continuum-valued functions,

(2) if Gn+1k+1 is connected, then G(Lk)

T≈ Gn+1

k is connected, and

(3) if L−1k is a union of continuum-valued functions, then f−1

k is a unionof continuum-valued functions.

19

Proof. (1) Let zk ∈ Lk(zk+1, . . . , zn+1). Since zk ∈ fk(zk+1) and fk is aunion of continuum-valued functions, there exists a continuum-valued func-tion g:Xk+1 → Xk such that zk ∈ g(zk+1) and G(g) ⊂ G(fk).

Let `:Gn+1k+1 → Xk be the set-valued function defined by `(xk+1, . . . , xn+1) =

g(xk+1). We see that zk ∈ `(zk+1, . . . , zn+1), and G(`) ⊂ G(Lk). Since g iscontinuum-valued, it follows that ` is continuum-valued. So, Lk:G

n+1k+1 → Xk

is a union of continuum-valued functions.

(2) Let A and B be closed sets such that A∪B = Gn+1k . Let ρ:Gn+1

k → Gk+1k

be projection. Since G(fk) is connected and ρ(A) ∪ ρ(B) = Gk+1k = G(f−1

k ),there is a point (pk, pk+1) ∈ ρ(A) ∩ ρ(B). Let (pk, pk+1, ak+2, . . . , an+1) ∈ A,and (pk, pk+1, bk+2, . . . , bn+1) ∈ B. We define the continuum-valued functionsg:Xk+1 → Xk and `:Gn+1

k+1 → Xk with pk ∈ g(pk+1) analogously as in (1).Recall that G(`) ⊂ G(Lk). Since Gn+1

k+1 is connected by hypothesis, it followsfrom Lemma 1 that G(`) is connected. Also, G(`) intersects both A and B;so, A and B are not mutually separated. Hence, Gn+1

k is connected.

(3) Let zk ∈ fk(zk+1). Let (zk+1, . . . , zn+1) ∈ Gn+1k+1 . Then (zk+1, . . . , zn+1) ∈

L−1k (zk), and, by assumption, there exists a continuum-valued g:Xk → Gn+1

k+1

such that (zk+1, . . . , zn+1) ∈ g(zk) and G(g) ⊂ G(L−1k ). Let g′:Xk → Xk+1

be defined by g′ = πk+1 ◦ g. We note that g′ is continuum-valued, since g iscontinuum-valued and πk+1 is a mapping. Also, zk+1 ∈ g′(zk) and G(g′) ⊂G(f−1

k ). So, f−1k is a union of continuum-valued functions.

Our remaining lemma, theorem, and corollaries give additional conditionsthat will ensure connectedness of partial graphs and inverse limits that havedetermining sequences in their associated inverse sequences.

Our most general results, in this setting, are those that follow.

Theorem 5. Suppose that {Xi, fi}ni=1 is a finite mixed inverse sequence oncontinua containing a sequence of max determining sequences {fji , . . . , fki}mi=1,where fj1 , . . . , fk1 is the first max determining sequence in f1, . . . , fn, for each1 ≤ i < m, fji , . . . , fki and fji+1

, . . . , fki+1are consecutive, and if km < n,

there are no determining sequences in fkm+1, . . . , fn. Suppose also that foreach 1 ≤ i ≤ m,

Fji,ki :Xki+1 → Gkiji

is a union, and G(Fji,ki) is connected.

Then Gn+11 is connected.

20

Proof. In order to simplify notation, we prove this theorem for m = 2, andwe denote the two max determining sequences by fj, . . . , fk and f`, . . . , fmwith k < `. The idea of the proof is straightforward using the tools we haveset up, and it will be clear that the proof will work analogously for m maxdetermining sequences when m > 2.

We consider the finite inverse sequence in our hypothesis and an amal-gamated form of it below. By Remark 3, the two inverse sequences havehomeomorphic partial graphs.

(1) X1f1←− . . .

fj−1←− Xjfj←− . . .

fk←− Xk+1fk+1←−

. . .f`−1←− X`

f`←− . . .fm←− Xm+1

fm+1←− . . .fn←− Xn+1

In sequence (1), by Observations 3 and 4, for 1 ≤ i < j and k+1 ≤ i < `,fi is a union. Amalgamating the max determining sequences, we get theinverse sequence below.

(2) X1f1←− . . .

fj−1←− Xj

πj |Gkj←− Gk

j

Fj,k←− Xk+1fk+1←−

. . .f`−1←− X`

π`|Gm`←− Gm

`

F`,m←− Xm+1fm+1←− . . .

fn←− Xn+1

In sequence (2), since πj and π` are mappings, we view them as degeneratecontinuum-valued functions. By hypothesis, the set-valued functions Fj,k andF`,m are unions of continuum-valued functions, and have connected graphs.In sequence (2), we have that all bonding functions through F`,m are unionswith connected graphs. If m = n, then, by Corollary 1, the partial graph ofinverse sequence (2) is connected. Hence, Gn+1

1 is connected. If m < n, thenthere are no determining sequences in fm+1, . . . , fn, so we have that either

(i) fi is an inverse union for all m+ 1 ≤ i ≤ n,

(ii) fi is a union for all m+ 1 ≤ i ≤ n , or

(iii) there exists m+ 1 < n′ < n such that fi a union for all m+ 1 ≤ i < n′,and fi is an inverse union for all n′ ≤ i ≤ n.

If (ii) is the case, it follows from Corollary 1 that the partial graph ofinverse sequence (2) is connected. Hence, Gn+1

1 is connected.If either (i) or (iii) is the case, it follows from Corollary 3 that the partial

graph of inverse sequence (2) is connected. Hence, Gn+11 is connected.

Corollary 4. Suppose that {Xi, fi}i≥1 is a mixed inverse sequence on con-tinua. Suppose also that, for each max determining sequence fj, . . . , fk,

21

Fj,k:Xk+1 → Gkj is a union, and G(Fj,k) is connected. Then lim

←−{Xi, fi}

is a continuum.

Proof. Case 1. Suppose that {Xi, fi}i≥1 has finitely many max determiningsequences. Pick n large enough so that all max determining sequences arecontained in the finite inverse sequence {Xi, fi}ni=1. It follows from Theorem5 that Gi+1

1 is connected for each i ≥ n. By Remark 2, Gi1 is also connected

for each 1 ≤ i ≤ n. As we saw in the proof of Corollary 2, it follows fromRemark 2 that lim

←−{Xi, fi} is a continuum.

Case 2. Suppose that {Xi, fi}i≥1 has infinitely many max determining se-quences. Pick the sequence {ni}i≥1 of integers, where for each i ≥ 1, fni

isthe last member of the nth

i max determining sequence. Then it follows fromTheorem 5 that Gni+1

1 is connected for each i ≥ 1, and as in Case 1, it followsthat lim

←−{Xi, fi} is a continuum.

Corollary 5. Suppose that {Xi, fi}i≥1 is an eventually alternating (beginningat k) inverse sequence on continua that contains no determining sequencesin f1, . . . , fk−1. Suppose also that, for each i ≥ k, Fi,i+1:Xi+2 → Gi+1

i is aunion, and G(Fi,i+1) is connected. Then lim

←−{Xi, fi} is a continuum.

Although Corollaries 4 and 5 reduce the determination of connectednessof an inverse limit of a mixed inverse sequence from checking that all par-tial graphs Gn

1 are connected to checking if, in max determining sequencesfj, . . . , fk, two conditions are present, it would be helpful to know of simple,observable properties of either Fj,k or some of the functions fi in the max de-termining sequence that would ensure that Fj,k is a union with a connectedgraph. Even though fk is a union, no simple property is apparent to theauthor that would make Fj,k a union. However, if Fj,k is found to be a union,

there are a few relatively easy properties that will make G(Fj,k)T≈ Gk+1

j

connected. They are

(A) Fj,k is continuum-valued,

(B) C(Fj,k) has a universal member,

(C) there exists a continuum K in G(Fj,k) such that whenever g ∈ C(Fj,k),K ∩G(g) 6= ∅, and

(D) fk:Xk+1 → Xk is full-valued at some point.

22

It is clear that each of properties (A), (B), and (C) make G(Fj,k) con-nected. In general, property (D) would be the easiest to check. Lemma 6below shows that property (D) ensures that G(Fj,k) is connected through amax determining sequence.

Lemma 6. Suppose that {Xi, fi}ni=1 is a finite mixed inverse sequence oncontinua containing a max determining sequence {fj, . . . , fk}. Suppose alsothat Fj,k is a union, and fk is full-valued at some point of Xk+1. Then G(Fj,k)is connected.

Proof. Since fk is full-valued at some point of Xk+1, there exists a point zk+1

in Xk+1 where fk(zk+1) = Xk. Let K = {zk+1} × Gkj . We claim that K is

a subcontinuum of G(Fj,k) that meets the graph of each g ∈ C(Fj,k). Since,for each j ≤ i ≤ k − 1, fi is an inverse union, we have that Gk

j is connected.So, K is connected.

If (zk+1, xj, . . . , xk) ∈ K, we have that xk ∈ fk(zk+1), and since (xj, . . . , xk)is inGk

j , it follows that (xj, . . . , xk, zk+1) ∈ Gk+1j . So, by definition, (xj, . . . , xk)

is in Fj,k(zk+1). Hence, K is a subcontinuum of G(Fj,k).Lastly, let g ∈ C(Fj,k). Since g is max continuum-valued, it follows that

g(zk+1) = Gkj . So, K ⊂ G(g). Hence, property (C) above is satisfied, and it

follows that G(Fj,k) is connected.

Corollary 6. Suppose that {Xi, fi}ni=1 is a finite mixed inverse sequence oncontinua containing a sequence of consecutive max determining sequences{fji , . . . , fki}mi=1 as in Theorem 5. Suppose also that for each 1 ≤ i ≤ m,

Fji,ki :Xki+1 → Gkiji

is a union, and one of properties (A)− (D) holds.

Then Gn+11 is connected.

Corollary 7. Suppose that {Xi, fi}i≥1 is a mixed inverse sequence on con-tinua. Suppose also that, for each max determining sequence fj, . . . , fk,Fj,k:Xk+1 → Gk

j is a union, and one of properties (A)-(D) holds. Thenlim←−{Xi, fi} is a continuum.

Acknowledgement. The author acknowledges correspondence, related tomixed inverse sequences, with Tom Ingram during the last few years.

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