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@ Appl. Gen. Topol. 19, no. 1 (2018), 9-20doi:10.4995/agt.2018.7061
c© AGT, UPV, 2018

Topological n-cells and Hilbert cubes in

inverse limits

Leonard R. Rubin

Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019, USA (lrubin@ou.edu)

Communicated by E. A. Sánchez-Pérez

Abstract

It has been shown by S. Mardešić that if a compact metrizable space X
has dim X ≥ 1 and X is the inverse limit of an inverse sequence of com-
pact triangulated polyhedra with simplicial bonding maps, then X must
contain an arc. We are going to prove that if X = (|Ka|, p

b

a, (A, �))
is an inverse system in set theory of triangulated polyhedra |Ka| with
simplicial bonding functions pba and X = lim X, then there exists a
uniquely determined sub-inverse system XX = (|La|, p

b

a

∣

∣|Lb|, (A, �)) of

X where for each a, La is a subcomplex of Ka, each p
b

a

∣

∣|Lb| : |Lb| → |La|
is surjective, and lim XX = X. We shall use this to generalize the
Mardešić result by characterizing when the inverse limit of an inverse
sequence of triangulated polyhedra with simplicial bonding maps must
contain a topological n-cell and do the same in the case of an inverse
system of finite triangulated polyhedra with simplicial bonding maps.
We shall also characterize when the inverse limit of an inverse sequence
of triangulated polyhedra with simplicial bonding maps must contain
an embedded copy of the Hilbert cube. In each of the above settings, all
the polyhedra have the weak topology or all have the metric topology
(these topologies being identical when the polyhedra are finite).

2010 MSC: 54B35; 54C25; 54F45.

Keywords: Hilbert cube; inverse limit; inverse sequence; inverse system;
polyhedron; simplicial inverse system; simplicial map; topologi-
cal n-cell; triangulation.

Received 29 December 2016 – Accepted 01 September 2017

http://dx.doi.org/10.4995/agt.2018.7061


L. R. Rubin

1. Introduction

Theorem 4.10.10 of [10] reads as follows.

Theorem 1.1. Every completely metrizable space X is homeomorphic to the
inverse limit of an inverse sequence (|Ki|m, p

i+1
i ) of metric polyhedra and PL

maps such that each Ki is locally finite-dimensional, card Ki ≤ wt X, and
each bonding map pi+1i : |Ki+1|m → |Ki|m is simplicial for some admissible

subdivision K′i of Ki, where admissibility guarantees the continuity of p
i+1
i :

|Ki+1|m → |Ki|m.

The notion of locally finite-dimensional used in Theorem 1.1 goes this way.
Let K be a simplicial complex. Whenever v is a vertex of K, then st(v, K) will
be the closed star of v in K, which is the subcomplex of K consisting of the
simplexes of K having v as a vertex and all faces of such simplexes. Then K
is called locally finite-dimensional if dim(st(v, K)) < ∞ for each v ∈ K(0).

One might wonder if an inverse sequence such as that in Theorem 1.1 could
be designed so that all the bonding maps1 are simplicial with respect to the
given triangulations; unfortunately this is not the case. It was shown by S.
Mardešić in Theorem 2.1 of [7], that if a compact metrizable space X has
dim X ≥ 1 and X is the inverse limit of an inverse sequence of compact trian-
gulated polyhedra with simplicial bonding maps, then X must contain an arc.
Since pseudo-arcs (see [8]) are metrizable compacta with dim ≥ 1 that contain
no arcs, then he was able to obtain Corollary 2.2 of [7], which says that there
exist metrizable compacta that cannot be written as the limit of an inverse se-
quence of compact triangulated polyhedra with simplicial bonding maps. The
proof of Theorem 2.1 of [7] is given without the assumption that the bonding
maps are surjective, but if they were, then by an observation of M. Levin, its
proof would be trivial.

The question of whether a given metrizable compactum could be written
as the limit of an inverse sequence of compact triangulated polyhedra with
simplicial bonding maps arose from our research in [9]. There we were able to
find, for the sake of extension theory, a “substitute” Z for any given compact
metrizable space X. This metrizable compactum Z is represented as the limit
of an inverse sequence of finite triangulated polyhedra in such a manner that
all the bonding maps are simplicial with respect to these triangulations. Since
the process of determining such a Z was complex, we were concerned to know
if it was necessary, that is, could we represent the given X “simplicially” from
the outset; the result of [7] made it apparent that we could not escape such a
complication.

We shall demonstrate, Proposition 2.7, that if X = (|Ka|, p
b
a, (A, �)) is an

inverse system in set theory of triangulated polyhedra |Ka| with simplicial
bonding functions pba, and X = lim X, then there exists a uniquely determined
sub-inverse system XX = (|La|, p

b
a

∣

∣|Lb|, (A, �)) of X where for each a, La is a

subcomplex of Ka, each p
b
a

∣

∣|Lb| : |Lb| → |La| is surjective, and lim XX = X.

1In this paper map means continuous function.

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Topological n-cells and Hilbert cubes in inverse limits

Hence for such a “simplicial” inverse system in which the polyhedra |Ka| are
given either the CW (weak) topology or the metric topology m, one may as
well assume for topological purposes that the bonding functions are surjective
maps.

In Corollary 3.3 we will characterize when the limit of an inverse sequence of
triangulated polyhedra with simplicial bonding maps must contain a topological
n-cell. In Proposition 3.5, we display a similar characterization in case we
are dealing with an inverse system of finite polyhedra and simplicial bonding
maps. Our Theorem 4.13 characterizes when the limit of an inverse sequence of
triangulated polyhedra and simplicial bonding maps must contain a copy of the
Hilbert cube I∞. We were not successful in obtaining such a result for inverse
systems even in the case that the coordinate spaces are finite polyhedra. In
Section 5 we shall provide what we could do for such systems.

2. Simplicial Inverse Systems

Let K be a simplicial complex. Then by |K|CW we mean the polyhedron |K|
with the CW-topology (sometimes called the weak topology) and by |K|m we
mean |K| with the metric topology m.2 If K is finite, then the CW-topology
is the same as the metric topology m, so we usually just write |K| with no
subscript. In case L is a simplicial complex and f : K → L is a simplicial
function, then f induces a function |f| : |K| → |L| which we say is simplicial
from |K| to |L|. In this setting we usually just write f instead of |f|; moreover,
one has that both f : |K|CW → |L|CW and f : |K|m → |L|m are maps.

We shall be concerned with inverse systems X = (Xa, p
b
a, (A, �)) with a

directed set (A, �) as indexing set. If X = lim X, then pa : X → Xa will denote
the a-coordinate projection. For x ∈ X, we shall typically write pa(x) = xa,
and denote x = (xa)a∈A or just x = (xa). If for each a ∈ A, Ya ⊂ Xa and
whenever a � b, pba(Yb) ⊂ Ya, then we call Y = (Ya, p

b
a|Yb, (A, �)) a sub-inverse

system of X. Clearly lim Y ⊂ lim X. In case (A, �) is (N, ≤), we simply denote
the inverse system X = (Xi, p

i+1
i ) and call it an inverse sequence.

The main result of this section is Proposition 2.7. It shows that if X is the
inverse limit of an inverse system in set theory of triangulated polyhedra and
simplicial maps, then there is a sub-inverse system consisting of subpolyhedra
determined by subcomplexes of the given triangulations such that the limit of
this sub-inverse system is X and that the restricted, and hence simplicial, maps
are surjective.

Definition 2.1. Let X = (|Ka|, p
b
a, (A, �)) be an inverse system in set theory

of triangulated polyhedra and simplicial bonding functions pba. We shall refer
to X as a simplicial inverse system. In case all |Ka| have the topology
CW or all have the topology m, then we shall denote all |Ka| respectively as
|Ka|CW or |Ka|m, and understand that all the functions p

b
a in set theory are

simultaneously maps. If all the functions pba are surjective, then we shall call

2One may consult [10] for more information about polyhedra.

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 11



L. R. Rubin

X a surjective inverse system. Let X = lim X, x ∈ X, and for each a ∈ A,
denote by σx,a the unique simplex of Ka with xa ∈ int σx,a.

Lemma 2.2. Let X = (|Ka|, p
b
a, (A, �)) be a simplicial inverse system, X =

lim X, and x ∈ X. Then the trace {σx,a | a ∈ A} of x in X has the property
that whenever a � b, pba(σx,b) = σx,a. Hence Xx = (σx,a, p

b
a|σx,b, (A, �)) is a

surjective simplicial sub-inverse system of X with bonding functions that are

simultaneously maps. Moreover, x ∈ lim Xx ⊂ X.

Definition 2.3. We shall refer to the uniquely determined inverse system
Xx = (σx,a, p

b
a|σx,b, (A, �)) of Lemma 2.2 as the trace of x in X.

Definition 2.4. Let X = (|Ka|, p
b
a, (A, �)) be a simplicial inverse system,

X = lim X, Q ⊂ X, for each a ∈ A denote MQ,a = {σy,a | y ∈ Q}, and define
LQ,a to be the collection of faces of elements of MQ,a.

Lemma 2.5. Let X = (|Ka|, p
b
a, (A, �)) be a simplicial inverse system, X =

lim X, and Q ⊂ X. Then for each a ∈ A:

(1) LQ,a is a uniquely determined subcomplex of Ka,
(2) if n ∈ N, and for all y ∈ Q, dim σy,a ≤ n, then dim LQ,a ≤ n, and
(3) if b ∈ A and a � b, pba(|LQ,b|) = |LQ,a|.

Hence XQ = (|LQ,a|, p
b
a

∣

∣|LQ,b|, (A, �)), which is uniquely determined by Q, is
a surjective simplicial sub-inverse system of X. Moreover, for each x ∈ Q, Xx
(see Lemma 2.2) is a sub-inverse system of XQ with x ∈ lim Xx, so Q ⊂ lim XQ.

Proof. Parts (1) and (2) are obviously true. To obtain (3), suppose that a � b.
First we show that pba(|LQ,b|) ⊂ |LQ,a|. Suppose that τ ∈ LQ,b, i.e., τ is a face of
an element σy,b ∈ MQ,b. Then Lemma 2.2 shows that p

b
a(σy,b) = σy,a ∈ MQ,a.

Since pba(τ) is a face of σy,a, then p
b
a(τ) ∈ LQ,a, so p

b
a(τ) ⊂ |LQ,a|. Now we

show the opposite inclusion, |LQ,a| ⊂ p
b
a(|LQ,b|). Suppose that τ ∈ LQ,a. Then

there exists y ∈ Q such that τ is a face of σy,a. As before, we know that
pba(σy,b) = σy,a; hence τ ⊂ σy,a = p

b
a(σy,b) ⊂ p

b
a(|LQ,b|), which proves the

desired inclusion. �

Definition 2.6. Let X = (|Ka|, p
b
a, (A, �)) be a simplicial inverse system,

X = lim X, and Q ⊂ X. Then we shall refer to the uniquely determined
inverse system XQ of Lemma 2.5 as the trace of Q in X.

Applying Lemmas 2.5 and 2.2, one arrives at the next result.

Proposition 2.7. If X = (|Ka|, p
b
a, (A, �)) is a simplicial inverse system and

X = lim X, then XX, the trace of X in X, is a surjective simplicial sub-inverse
system of X with lim XX = X. This shows that X can be represented as the
limit of a surjective simplicial inverse system.

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Topological n-cells and Hilbert cubes in inverse limits

3. Topological Cells in Inverse Limits

In Corollary 3.3 we shall characterize the conditions under which the inverse
limit of a simplicial inverse sequence contains a topological n-cell. The same
will be done in Corollary 3.5 for a simplicial inverse system in which the coor-
dinate spaces are finite polyhedra. For inverse sequences, we will make use of
the class of stratifiable spaces; such spaces are convenient for applications when
considering limits of inverse sequences. An exposition of generalized metrizable
spaces, including stratifiable spaces, is given by G. Gruenhage [3] in the Hand-
book of Set-Theoretic Topology. In that work, it is assumed that all spaces
under consideration are T1 and regular. But for our purposes, we will only
require that they be T1.

We note that stratifiable spaces were first called M3-spaces, but the term
stratifiable was introduced in [1] and this nomenclature became standard thence-
forward. Lemma 3.1 contains a list of properties of stratifiable spaces. Let us
first see which ones can be verified by reference to page numbers from [3]. The
definition is given on page 426; we shall not repeat it here. Using that defini-
tion and the T1 property, it is easy to prove that stratifiable spaces are regular;
hence they are Hausdorff. Theorem 5.7 on page 457 gives us paracompactness,
and Theorem 5.10 on page 458 shows that they are hereditarily stratifiable and
countably productive. Hence the limit of an inverse sequence of stratifiable
spaces is stratifiable. Corollary 5.12(ii) on page 459 gives us that metrizable
spaces are stratifiable. So the only statements in Lemma 3.1 yet to be verified
are the one in (4) concerning |K|CW, (5), and (7). We need to get at these
from other references.

Every polyhedron |K|CW has the structure of a CW-complex. If one views
the Introduction of [2] (see Corollary 8.6), one can see that all CW-complexes
and hence all polyhedra are stratifiable spaces. This gives us the first part of
(4). The main result of [5] (see also [6]) shows that covering dimension dim is
preserved in the inverse limit of an inverse sequence of stratifiable spaces, so
(7) is established. We get (5) from Theorem 3.6 of [4].

Lemma 3.1. The following are some facts about stratifiable spaces.

(1) Every stratifiable space is paracompact and Hausdorff.
(2) Every subspace of a stratifiable space is stratifiable.
(3) All metrizable spaces are stratifiable.
(4) For each simplicial complex K, both |K|CW and |K|m are stratifiable.
(5) If Y ⊂ X and X is a stratifiable space, then dim Y ≤ dim X.
(6) The limit of an inverse sequence of stratifiable spaces is stratifiable.
(7) If X = (Xi, p

i+1
i ) is an inverse sequence of stratifiable spaces, X =

lim X, n ≥ 0, and for each i, dim Xi ≤ n, then dim X ≤ n.

Proposition 3.2. Let X = (|Ki|CW, p
i+1
i ) be a simplicial inverse sequence,

X = lim X, and n ∈ N. If dim X ≥ n, then there exist i0 ∈ N and a sequence
(τi)i≥i0 such that for each i ≥ i0, τi is an n-simplex of Ki and p

i+1
i carries τi+1

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 13



L. R. Rubin

topologically onto τi. The same is true if we replace the topology CW, where it
appears above, by the metric topology m.

Proof. Applying Proposition 2.7, there is no loss of generality in assuming
that pi+1i is surjective for all i. Using Lemma 3.1(4,6), one sees that X is
stratifiable. An application of Lemma 3.1(7) shows this: if it is true that for
all i, dim |Ki| < n, then one would have that dim X < n. So there is a first
i0 ∈ N with dim Ki0 ≥ n. Let τi0 be a simplex of Ki0 with dim τi0 = n. Using
the fact that for each i ≥ i0, p

i+1
i is simplicial and surjective, one can choose a

sequence (τi)i≥i0 as requested. The same argument can be applied if we replace
the topology CW, where it appears, by the metric topology m. �

We obtain a corollary to Lemma 3.1(4,5) and Proposition 3.2.

Corollary 3.3. Let X = (|Ki|CW, p
i+1
i ) be a simplicial inverse sequence, X =

lim X, and n ∈ N. Then X contains a topological n-cell if and only if dim X ≥
n. The same is true if we replace the topology CW, where it appears, by the
metric topology m.

Proposition 3.4. Let X = (|Ka|, p
b
a, (A, �)) be a simplicial inverse system

where all the |Ka| are finite polyhedra, X = lim X, and n ∈ N. If dim X ≥ n,
then there exists d ∈ A such that for each a ∈ A with d � a, there is an n-
simplex τa of Ka such that if b ∈ A with a � b, then p

b
a carries τb topologically

onto τa. Thus, X contains a topological n-cell.

Proof. We may assume that (A, �) has no upper bound. Applying Proposition
2.7, there is no loss of generality in assuming that pba is surjective for all a � b.
It is moreover true that X is a compact Hausdorff space. Since dim X ≥ n,
there has to be a cofinal subset A0 of A such that dim Ka ≥ n for all a ∈ A0.
We may as well require that A has this property from the outset. Fix d ∈ A.
Then the set of a ∈ A with d � a is cofinal in A, so we shall assume that for
all a ∈ A, d � a.

Now fix an n-simplex τd in Kd, let xτd ∈ int τd, and Hd = {xτd}. For each
a ∈ A, there is at least one n-simplex τ ∈ Ka such that p

a
d
(τ) = τd. Let Fa

be the collection of such n-simplexes, and for each τ ∈ Fa, select the unique
element xτ ∈ int τ with p

a
d
(xτ ) = xτd. Denote Ha = {xτ | τ ∈ Fa}. Then for all

a ∈ A, Ha is a finite, nonempty subset of |Ka|, and if u ∈ Ha, then p
a
d(u) = xτd.

We claim that if a � b, then pba(Hb) ⊂ Ha. For let τ ∈ Fb; we must show
that pba(xτ ) ∈ Ha. Now p

a
d ◦ p

b
a(xτ ) = p

b
d(xτ ) = xτd. Also, p

b
d(τ) = τd. It

follows that τ∗ = pba(τ) is an n-simplex of Ka and p
a
d
(τ∗) = τd. Thus, τ

∗ ∈ Fa
and pba(xτ ) = xτ∗ ∈ Ha as required. From this we get a sub-inverse system
H = (Ha, p

b
a|Hb, (A �)) of X consisting of nonempty discrete finite sets Ha.

Thus lim H 6= ∅. Select y ∈ lim H ⊂ lim X. From Lemma 2.2, the trace
of y in X, Xy = (σy,a, p

b
a|σy,b, (A, �)) is a surjective simplicial sub-inverse

system of X. Since dim σy,a = n for all a, then each p
b
a|σy,b : σy,b → σy,a is a

homeomorphism. Clearly, lim Xy ⊂ lim X is a topological n-cell. �

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 14



Topological n-cells and Hilbert cubes in inverse limits

Corollary 3.5. Let X = (|Ka|, p
b
a, (A, �)) be a simplicial inverse system where

all the |Ka| are finite polyhedra, X = lim X, and n ∈ N. Then X contains a
topological n-cell if and only if dim X ≥ n.

4. Hilbert Cubes in Limits of Inverse Sequences

The main result of this section is Theorem 4.13. It characterizes when
the limit of a simplicial inverse sequence must contain a copy of the Hilbert
cube. First, let us review some concepts from dimension theory. Recall that an
infinite-dimensional space is called countable-dimensional if it can be written
as the union of subspaces Xn, n ∈ N, each Xn having dimension ≤ n. It is
called strongly countable-dimensional if it can be written as the union of closed
subspaces Xn, n ∈ N, each Xn having dimension ≤ n. Of course, strongly
countable-dimensional spaces are countable-dimensional.

From Corollaries 3.3 and 3.5, respectively, we get Propositions 4.1 and 4.2.

Proposition 4.1. Let X = (|Ki|CW, p
i+1
i

) be a simplicial inverse sequence and
X = lim X. If dim X = ∞, then X contains a strongly countable dimensional
subspace Y =

⋃

{Yi | i ∈ N} such that for each i, Yi is a topological i-cell. The
same is true if we replace the topology CW, where it appears above, by the
metric topology m.

Proposition 4.2. Let X = (|Ka|CW, p
b
a, (A, �)) be a simplicial inverse system

and X = lim X. If all the |Ka| are finite polyhedra and dim X = ∞, then X
contains a strongly countable dimensional subspace Y =

⋃

{Yi | i ∈ N} such that
for each i, Yi is a topological i-cell.

As usual, I = [0, 1], the unit interval. We shall denote the Hilbert cube as
I∞, that is, I∞ =

∏

{Ii | i ∈ N} where for each i, Ii = I. For each i ∈ N,
let pi+1i : I

i+1 → Ii be the i-coordinate projection. Remember that strongly
infinite-dimensional spaces are not countable-dimensional. Since I∞ is strongly
infinite-dimensional, it is not countable-dimensional. One may consult [10] for
more information on this subject.

Lemma 4.3. Let G = (Ii, pi+1i ) be the inverse sequence having the property

that for each i, pi+1
i

: Ii+1 → Ii is the coordinate projection. Then lim G ∼= I∞.

Proof. Since both I∞ and lim G are compact metrizable spaces, it is sufficient
to find a bijective map from I∞ to lim G. Define a map h : I∞ → lim G by
setting h(x1, x2, x3, . . . ) = (x1, (x1, x2), (x1, x2, x3), . . . ). Surely h is a map; we
leave it to the reader to show that h is a bijection. �

Whenever V is the vertex set of a simplex σ, then an arbitrary element x
of σ will be written x =

∑

{xvv | v ∈ V}, where for each v ∈ V, xv is the
v-barycentric coordinate of x.

Lemma 4.4. Let n ∈ N, σ be an n-simplex with vertex set V, τ0 an (n−1)-face
of σ, W the vertex set of τ0, v ∈ V \ W, and µ : σ → τ0 a simplicial retraction.
Then µ(u) = u for each u ∈ W, and there is a unique w ∈ W with µ(v) = w.

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L. R. Rubin

Indeed, if x =
∑

{xvv | v ∈ V} ∈ σ, then µ(x) =
∑

{buu | u ∈ W} ∈ τ0 where
bu = xv + xw if u = w, and bu = xu otherwise.

Lemma 4.5. Let n ∈ N, σ be an n-simplex, τ an (n − 1)-simplex, and q :
σ → τ a simplicial surjection. Then there exist a unique (n − 1)-face τ0 of σ,
a unique simplicial retraction µ : σ → τ0, and a unique simplicial isomorphism
q0 : τ0 → τ, such that q = q0 ◦ µ.

Lemma 4.6. Let n ∈ N, σ be an n-simplex, τ0 an (n − 1)-face of σ, and µ :
σ → τ0 a simplicial retraction of σ to τ0. Suppose that D ⊂ int τ0 is nonempty
and compact. Let V, W, v, and w come from Lemma 4.4. We claim that for
any neighborhood U of ∂σ in σ, there is an embedding H : D × I → U ∩ int σ
such that (µ| im(H)) ◦ H = p : D × I → D, where p : D × I → D is the
coordinate projection.

Proof. Let (x, t) ∈ D × I, x =
∑

{xuu | u ∈ W} ∈ D ⊂ int τ0. Define H(x, t) ∈
σ so that its v-barycentric coordinate is (1 − t)xw, its w-barycentric coordinate
is txw, and for any u ∈ V \ {v, w}, its u-barycentric coordinate is xu. Then
clearly H : D×I → σ is a map. To show that H is injective, let y =

∑

{yuu | u ∈
W} ∈ D, {t, s} ⊂ I, and (x, t) 6= (y, s). If u ∈ W \ {w}, and xu 6= yu, then
H(x, t) 6= H(y, s) independently of t and s. Hence we may as well assume
that xu = yu for all u ∈ W \ {w}. Suppose that H(x, t) = H(y, s). If t = s,
then x 6= y, that is, xw 6= yw. By the definition of H, (1 − t)xw = (1 − t)yw
and txw = tyw. Since one of {1 − t, t} does not equal 0, then xw = yw, a
contradiction. Hence t 6= s. Since D ⊂ int τ0, then xw 6= 0, so txw 6= sxw. This
implies that t = s, another contradiction. Therefore H(x, t) 6= H(y, s). We
have demonstrated that H is injective which shows that H is an embedding
because of compactness. One easily checks that (µ| im(H))◦H = p : D×I → D.

Notice that for x ∈ D ⊂ int τ0 as above, for all u ∈ W, xu > 0. This is true
in particular if u = w. If t /∈ {0, 1}, both (1 − t)xw > 0 and txw > 0. Hence
for all u ∈ W, the u-barycentric coordinates of H(x, t) are > 0. Therefore
if 0 < a < b < 1 and we restrict H to D × [a, b], we get an embedding of
D × [a, b] into int σ. But, the v-barycentric coordinate of H(x, 1) equals 0. So
H(D × {1}) ⊂ ∂σ. Taking a sufficiently close to 1, we get that H(D × [a, b]) ⊂
U ∩ int σ. It is now simply a matter of reparameterizing [a, b] so that it is
replaced by [0, 1], and we have our proof. �

Lemma 4.7. Let n ∈ N, σ be an n-simplex, τ an (n−1)-simplex, and q : σ → τ
a simplicial surjection. Suppose that E is a nonempty compact subset of int τ.
Then for any neighborhood U of ∂σ in σ, there is an embedding H∗ : E × I →
U ∩ int σ such that (q| im(H∗)) ◦ H∗ = p : E × I → E, where p : E × I → E is
the coordinate projection.

Proof. Apply Lemma 4.5 to q : σ → τ. Let τ0 be the unique (n − 1)-face of
σ, µ : σ → τ0 the unique simplicial retraction, and q0 : τ0 → τ the unique
simplicial isomorphism such that q = q0 ◦ µ. Put D = q

−1
0 (E) ⊂ int τ0. Apply

Lemma 4.6 to get an embedding H : D × I → U ∩ int σ having the property
that (µ| im(H)) ◦ H = p : D × I → D, where p : D × I → D is the coordinate

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Topological n-cells and Hilbert cubes in inverse limits

projection. Define H∗ : E × I → int σ by H∗(e, t) = H(q−10 (e), t). Surely H
∗

is an embedding of E × I into U ∩ int σ. Suppose that (e, t) ∈ E × I. Then
q ◦ H∗(e, t) = q0 ◦ µ ◦ H

∗(e, t) = q0 ◦ µ ◦ H(q
−1
0 (e), t) = q0 ◦ p(q

−1
0 (e), t) =

q0 ◦ q
−1
0 (e) = e. �

Lemma 4.8. Let m < n ∈ N and {σi | 0 ≤ i ≤ n − m} be a set such that for
each 0 ≤ i ≤ n − m, σi is an (m + i)-simplex. For each 1 ≤ i ≤ n − m, let
qi : σi → σi−1 be a simplicial surjection and put q = q1◦· · ·◦qn−m : σn−m → σ0.
Let E be a nonempty compact subset of int σ0 and U a neighborhood of ∂σn−m
in σn−m. Then there is an embedding H

∗ : E × In−m → U ∩ int σn−m such
that (q| im(H∗)) ◦ H∗ = p : E × In−m → E, where p : E × In−m → E is the
coordinate projection.

Proof. An application of Lemma 4.7 shows that this result is true in every case
where n − m = 1. Suppose that k ∈ N, and the lemma is true in every case
where n − m = k. Now assume that n − m = k + 1 and we are given the
above data, only this time with one more map in the composition. Note that
in this setting, q = q′ ◦ qk+1 where qk+1 : σk+1 → σk, dim σk+1 = dim σk + 1,
q′ = q1 ◦· · ·◦qk : σk → σ0, and k = n−(m+1) > 0. Also, U is a neighborhood
of ∂σk+1 in σk+1. Thus, m + 1 < n, so we may apply the inductive hypothesis
to the map q′. This gives us an embedding H : E × Ik → int σk such that
(q′| im(H)) ◦ H = p′ : E × Ik → E, where p′ : E × Ik → E is the coordinate
projection.

We now have the nonempty compact subset im H ⊂ int σk and of course
k + 1 − k = 1. So we may apply the fact that our result is true for n = k + 1,
m = k. This gives us an embedding H′ : (im H)× I into U ∩ int σk+1 such that
(qk+1| im(H

′)) ◦ H′ = p∗ : (im H) × I → im H, where p∗ : (im H) × I → im H
is the coordinate projection. Define H∗ : E × Ik × I → U ∩ int σk+1 by
H∗(e, s, t) = H′(H(e, s), t). It follows that H∗ is an embedding.

We must prove that (q| im(H∗)) ◦ H∗ = p : E × Ik+1 → E, where p :
E × Ik+1 → E is the coordinate projection. Let (e, s, t) ∈ E × Ik × I. Then
q◦H∗(e, s, t) = q◦H′(H(e, s), t) = q′◦qk+1◦H

′(H(e, s), t) = q′◦p∗(H(e, s), t) =
q′ ◦ H(e, s) = p′(e, s) = e. Our proof is complete. �

Applying Lemmas 4.5 and 4.8, one obtains a corollary.

Corollary 4.9. Let σ and τ be simplexes such that dim τ = m < dim σ = n,
suppose that p : σ → τ is a simplicial surjection, E is a compact subset of
int τ and U is a neighborhood of bd σ in σ. Then there exists an embedding
H : E × In−m → U ∩ int σ such that p ◦ H : E × In−m → τ is the coordinate
projection E × In−m to E.

Proposition 4.10. Suppose that S = (σi, q
i+1
i ) is a surjective simplicial in-

verse sequence such that for each i, σi is an i-simplex. Then lim S contains an
embedded copy of I∞.

Proof. Let E ⊂ int σ1 be a closed interval, and identify E with I. Apply
Lemma 4.7 in such a way that I × I ⊂ int σ2 and q

2
1|I × I : I × I → I ⊂ int σ1

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 17



L. R. Rubin

is the coordinate projection (t1, t2) 7→ t1. Next apply Lemma 4.7 again in such
a way that I2 ×I ⊂ int σ3 and q

3
2|I

2 ×I : I2 ×I → I2 ⊂ int σ2 is the coordinate
projection (t1, t2, t3) 7→ (t1, t2).

Continuing recursively in this manner, we land up with a sub-inverse se-
quence of S of the form G = (Ii, pi+1i ) from Lemma 4.3. Therefore lim G

∼=
I∞ ⊂ lim S as requested. �

Corollary 4.11. Suppose that S = (σi, q
i+1
i ) is a surjective simplicial inverse

sequence such that for each i, σi is a simplex, and there exists an increasing
sequence (ni) in N such that for each i, dim σni < dim σni+1. Then lim S
contains an embedded copy of I∞.

Proof. Since the sequence (ni) is increasing, we may replace S with the inverse
sequence (σni, q

ni+1
ni ) whose inverse limit is homeomorphic to lim S. To conserve

notation, let us assume that the given inverse sequence S = (σi, q
i+1
i ) already

has the property that dim σi < dim σi+1 for all i. One may also assume that
1 ≤ dim σ1. Select a 1-face τ1 of σ1. Choose a 2-face τ2 of σ2 with q

2
1(τ2) = τ1.

Similarly, choose a 3-face τ3 of σ3 with q
3
2(τ3) = τ2. This process can be

continued recursively so that we end up with a sequence (τi) having the property
that for each i, dim τi = i, τi is a face of σi, and q

i+1
i

|τi+1 : τi+1 → τi is
a simplicial surjection. The surjective simplicial sub-inverse sequence S0 =
(τi, q

i+1
i |τi+1) of S replicates the inverse sequence in Proposition 4.10, so I

∞

embeds in lim S0 which in turn embeds in lim S. �

Lemma 4.12. Let X = (|Ki|CW, p
i+1
i ) be a simplicial inverse sequence, and

put X = lim X. Suppose that X contains a strongly infinite-dimensional sub-
space Q. Then there exist x ∈ Q and an increasing sequence (ni) in N, so that
the trace Xx of x in X has the property that for each i, dim σx,ni < dim σx,ni+1.
The same is true if we replace the topology CW, where it appears above, by the
metric topology m.

Proof. For each x ∈ Q ⊂ X, let Xx be the trace of x in X. Then for all
i, σx,i ∈ Ki and p

i+1
i (σx,i+1) = σx,i, so dim σx,i ≤ dim σx,i+1; moreover,

x ∈ lim Xx. Let us suppose, for the purpose of reaching a contradiction, that
for all x ∈ Q, there exists nx ∈ N such that dim σx,i ≤ nx for all i. For each
n ∈ N, let Qn = {x ∈ Q | nx ≤ n}. Then Q =

⋃

{Qn | n ∈ N}.
Fix n ∈ N, and for each i ∈ N, let MQn,i be as in Definition 2.4. Then all the

simplexes in MQn,i have dimension ≤ n. So by Lemma 2.5(2), dim LQn,i ≤ n.
Applying Proposition 2.7, we get the sub-inverse sequence XQn = (|LQn,i|,

pi+1i
∣

∣|LQn,i+1|) of X, with Qn ⊂ Xn = lim XQn. Surely Xn is a stratifiable
space and dim Xn ≤ n. Thus, dim(Qn ∩Xn) ≤ n. Hence Q =

⋃

{Qn ∩Xn | n ∈
N} is countable-dimensional, which is false. This same argument works if we
replace the topology CW, where it appears, by the metric topology m. Our
proof is complete. �

Putting together Corollary 4.11 and Lemma 4.12, we obtain a theorem.

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 18



Topological n-cells and Hilbert cubes in inverse limits

Theorem 4.13. Let X = (|Ki|CW, p
i+1
i ) be a simplicial inverse sequence, and

put X = lim X. Then X contains an embedded copy of I∞ if and only if there
is a collection {σi | i ∈ N} and an increasing sequence (ni) in N, such that for
each i,

(1) σi is a simplex of Ki,
(2) pi+1i (σi+1) = σi, and
(3) dim σni < dim σni+1.

The same is true if we replace the topology CW, where it appears above, by the
metric topology m.

5. Strongly Infinite Dimensional Sets in Limits of Inverse
Systems of Finite Polyhedra

We present a result for inverse systems of finite polyhedra that is parallel
to Lemma 4.12. We however do not have a result that is similar to that of
Theorem 4.13.

Proposition 5.1. Let X = (|Ka|, p
b
a, (A, �)) be a simplicial inverse system

where all the |Ka| are finite polyhedra, and let X = lim X. Suppose that X
contains a strongly infinite-dimensional closed subspace Q. Then there exists
x ∈ X (indeed, x ∈ Q) so that the trace Xx of x in X satisfies the property that
for each a ∈ A and n ∈ N, there exists a � b such that dim σx,b ≥ n. Hence
there exists a sequence (ai) in A such that for each i, ai � ai+1, ai 6= ai+1, and
dim σai < dim σai+1.

Proof. Since X contains a strongly infinite-dimensional closed subspace, then
(A, �) has no upper bound. For each x ∈ Q ⊂ X, let Xx be the trace of x
in X. Let us suppose, for the purpose of reaching a contradiction, that for all
x ∈ Q, there exist ax ∈ A and nx ∈ N such that for all ax � b, dim σx,b ≤ nx.
For each n ∈ N, let Qn = {x ∈ Q | nx ≤ n}. Then Q =

⋃

{Qn | n ∈ N}.
Fix n ∈ N, and for each a ∈ A, let MQn,a be as in Definition 2.4. Then

whenever ax � b, by Lemma 2.5(2), dim LQn,b ≤ n. One should note that
{b ∈ A | ax � b} is cofinal in A. Applying Definition 2.6, we get the sub-
inverse system XQn = (|LQn,a|, p

b
a

∣

∣|LQn,b|, (A �)) of X, with Qn ⊂ Xn =
lim XQn. Surely, Xn is a compact Hausdorff space and dim Xn ≤ n. Hence
Q =

⋃

{Qn ∩Xn | n ∈ N} is strongly countable-dimensional, which is false since
Q is strongly infinite-dimensional. Our proof is complete. �

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