nLab
tube lemma

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

The tube lemma may refer to one of several fundamental lemmas in topology (point-set topology to be more exact1) that underlie many arguments involving compact spaces. That is to say: much like the Yoneda lemma of pure category theory, there are various closely related statements that can be said to fall under the rubric “tube lemma”; we consider several examples here.

Statements and proofs

Proposition

(closed-projection characterization of compactness)

If XX is any space and YY is a compact, then the projection map p:X×YXp: X \times Y \to X out of the product topological space is a closed map, i.e., under direct image (the left adjoint p:P(X×Y)P(X)\exists_p: P (X \times Y) \to P(X) to the inverse image map p *:P(X)P(X×Y)p^\ast: P(X) \to P(X \times Y)), closed sets in X×YX \times Y are mapped to closed sets in XX.

Assuming the principle of excluded middle, the statement of prop. 1 is equivalent to an alternative formulation:

Proposition

let p:P(X×Y)P(X)\forall_p: P(X \times Y) \to P(X) be the right adjoint to p *:P(X)P(X×Y)p^\ast: P(X) \to P(X \times Y). (In classical logic, p=¬ p¬\forall_p = \neg \exists_p \neg.) Then

  • If YY is compact, then p:P(X×Y)P(X)\forall_p: P(X \times Y) \to P(X) takes open sets in X×YX \times Y to open sets in XX. (Compare overt space.)
Proof

This alternative may be proven directly as follows. Let WX×YW \subseteq X \times Y be an open set, and suppose {x} p(W)\{x\} \subseteq \forall_p(W). This means precisely that p *{x}={x}×YWp^\ast\{x\} = \{x\} \times Y \subseteq W. We are required to show that there is a neighborhood OO of xx such that O p(W)O \subseteq \forall_p(W), which is to say O×YWO \times Y \subseteq W.

This O×YO \times Y may be pictured as a little open “tube” around {x}×Y\{x\} \times Y which fits inside WW, thus explaining the name of the eponymous lemma.

Lemma

(Tube lemma)

Let XX be a topological space and let YY be a compact topological space. For a point xXx \in X and WW open in X×YX \times Y, if {x}×YW\{x\} \times Y \subseteq W, then there is an open neighborhood OO of xx such that the “tube” O×YO \times Y around {x}×Y\{x \} \times Y is still contained: O×YWO \times Y \subseteq W.

Proof

Let 𝒪 x\mathcal{O}_x denote the set of open neighborhoods of xx, and for U𝒪 xU \in \mathcal{O}_x, let UWU\multimap W be the largest open VV of YY such that U×VWU \times V \subseteq W. The collection {UW:U𝒪 x}\{U \multimap W: U \in \mathcal{O}_x\} is an open cover of YY; since YY is compact, there is a finite subcover U 1W,U 2W,,U nWU_1 \multimap W, U_2 \multimap W, \ldots, U_n \multimap W. Then O=U 1U n𝒪 xO = U_1 \cap \ldots \cap U_n \in \mathcal{O}_x is the desired open. For, OU iO \subseteq U_i implies (U iW)(OW)(U_i \multimap W) \subseteq (O \multimap W), so that

Y= i=1 n(U iW)OWY = \bigcup_{i=1}^n (U_i \multimap W) \subseteq O \multimap W

and this implies O×YO×(OW)WO \times Y \subseteq O \times (O \multimap W) \subseteq W.

Proof

(of the tube lemma 1 via closed-projection characterization of compactness)

Let

C(X×Y)\W C \coloneqq (X \times Y) \backslash W

be the complement of WW. Since this is closed, by prop. 1 also its projection p X(C)Xp_X(C) \subset X is closed.

Now

{x}×YW {x}×YC= {x}p X(C)= \begin{aligned} \{x\} \times Y \subset W & \;\Leftrightarrow\; \{x\} \times Y \; \cap \; C = \emptyset \\ & \;\Rightarrow\; \{x\} \cap p_X(C) = \emptyset \end{aligned}

and hence by the closure of p X(C)p_X(C) there is (by this lemma) an open neighbourhood U x{x}U_x \supset \{x\} with

U xp X(C)=. U_x \cap p_X(C) = \emptyset \,.

This means equivalently that U x×YC=U_x \times Y \cap C = \emptyset, hence that U x×YWU_x \times Y \subset W.

Consequences

Tychnoff theorem

Corollary

(binary Tychonoff theorem)

The product topological space X×YX \times Y of two compact topological spaces X,YX, Y is itself compact.

Proof

Let 𝒰\mathcal{U} be any open cover of X×YX \times Y, and let \mathcal{B} be the collection of opens of X×YX \times Y that are unions of finitely many elements of 𝒰\mathcal{U}. For each xXx \in X, we have that {x}×Y\{x\} \times Y is compact since YY is, so there is BB \in \mathcal{B} such that {x}×YB\{x\} \times Y \subseteq B, whence there is U𝒪 xU \in \mathcal{O}_x with U×YBU \times Y \subseteq B by the tube lemma.

It follows that the collection

{UopeninX:U×YBforsomeB}\{U\; open\; in\; X: U \times Y \subseteq B \; for\; some\; B \in \mathcal{B}\}

covers XX, whence by compactness of XX there is a finite subcover U 1,,U nU_1, \ldots, U_n for which U i×YB iU_i \times Y \subseteq B_i for some B iB_i \in \mathcal{B}, and then B=B 1B nB = B_1 \cup \ldots \cup B_n belongs to \mathcal{B} and is all of X×YX \times Y.

Exponentiability of the compact-open topology

The tube lemma may be used to show that the mapping space from a locally compact topological space to any topology spaces equipped with the compact-open topology is an exponential object in the category Top of topological spaces. See there.

Converse of the tube lemma

As explained above, the three statements

  1. Tube lemma, Lemma 1

  2. p:P(X×Y)P(X)\forall_p: P(X \times Y) \to P(X) preserves openness if XX is compact

  3. p:P(X×Y)P(X)\exists_p: P(X \times Y) \to P(X) preserves closedness if XX is compact (prop. 1)

are virtually tautologically equivalent, and in this sense any one of these forms may be referred to as the tube lemma. In this section, we indicate that the converse of the tube lemma holds, stated as follows.

Theorem

(closed-projection characterization of compactness) If for all spaces XX the projection p:X×YXp: X \times Y \to X is closed (form 3), then YY is compact.

Various proofs may be given. If one likes the characterization of compactness that says every ultrafilter converges, then a neat conceptual proof runs as follows: given a space YY satisfying the hypothesis, take XX to be the Stone-Cech compactification βY\beta Y of the discrete space Y dY_d on the underlying set of YY, and let CβY×YC \subseteq \beta Y \times Y be the convergence relation (i.e., (U,y)(U, y) belongs to CC iff the ultrafilter UU converges to yy). One checks that CC is closed. Then the direct image p(C)p(C) is closed by hypothesis. Notice also that p(C)p(C) contains the set of principal ultrafilters prin(y)prin(y) (prin(y)prin(y) converges to yy after all), and this set may be identified with the dense set Y dβYY_d \subset \beta Y. Being closed and dense, p(C)p(C) is all of βY\beta Y, but this simply says that every ultrafilter on YY converges, so that YY is compact.

The characterization of compactness via ultrafilter convergence has a slight drawback of relying on the ultrafilter theorem, a kind of choice principle. If one doesn’t like this, there are various workarounds that avoid it, but let it be said that all proofs (some of which are collected here) have basically the same intuitive character. Given YY, one forms a space XX by adjoining one or more ideal points to the discrete space Y dY_d, where the open sets around an ideal point correspond to elements of some filter FF that we want to show clusters around, or converges to, some point yy (in order to show compactness of YY). Just as for the convergence relation in the proof sketched above, one passes to the closure CC of the diagonal YY d×YX×YY \to Y_d \times Y \hookrightarrow X \times Y. The image p(C)p(C) is closed and contains the dense subset Y dY_d, and so contains any ideal point pp, and thus (p,y)C(p, y) \in C for some yy. This turns out to mean FF converges or clusters to yy, as desired.

Here is a more precise enactment of one such proof.

Proof

Let YY satisfy the hypothesis of Theorem 1; to prove YY is compact, suppose 𝒞\mathcal{C} is some collection of closed sets such that every finite intersection of elements of 𝒞\mathcal{C} is inhabited. Let X=Y d{}X = Y_d \sqcup \{\infty\}, formed by adjoining an ideal point \infty to the discrete space Y dY_d on the underlying set of YY, and stipulating that whenever K𝒞K \in \mathcal{C}, the set K{}K \sqcup \{\infty\} is an open neighborhood of \infty. The topology thus generated consists of arbitrary subsets UYU \subseteq Y together with sets F{}F \sqcup \{\infty\} where FF belongs to the filter generated by 𝒞\mathcal{C}. Note that the finite intersection property of 𝒞\mathcal{C} guarantees that the filter is proper, meaning in particular \infty is not an open point, equivalently that the closure of Y dY_d in XX is all of XX.

We have a diagonal embedding YΔY d×YX×YY \stackrel{\Delta}{\to} Y_d \times Y \hookrightarrow X \times Y; let CC be the closure of YY in X×YX \times Y. Of course p(C)Xp(C) \subseteq X contains Y dY_d, and is closed by hypothesis, so as we just observed, p(C)=Xp(C) = X. So p(C)\infty \in p(C); this means that (,y)C(\infty, y) \in C for some yYy \in Y. By closure of CC, for each K𝒞K \in \mathcal{C} the neighborhood (K{})×U(K \sqcup \{\infty\}) \times U of (,y)(\infty, y) meets Δ(Y)X×Y\Delta(Y) \subseteq X \times Y. This just says every open U𝒪 yU \in \mathcal{O}_y intersects KK; since KK is closed in YY, this means yKy \in K. Thus

y K𝒞Ky \in \bigcap_{K \in \mathcal{C}} K

and the non-emptiness of this intersection proves that YY is compact.

References


  1. The use of the phrase “point-set topology” signals that the tube lemma proper does refer to points, as opposed to the “point-free” or “pointless” topology as developed in the theory of locales. Point-free analogues of the main consequences of the tube lemma, such as the fact that the product of two compact topological spaces is again compact, are quite nontrivial and require a significantly different approach.

Last revised on May 11, 2017 at 11:18:25. See the history of this page for a list of all contributions to it.