nLab
closed subspace

This entry is about closed subsets of a topological space. For other notions of “closed space” see for instance closed manifold.


Context

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Contents

Idea

A subset CC of a topological space (or more generally a convergence space) XX is closed if its complement is an open subset, or equivalently if it contains all its limit points. When equipped with the subspace topology, we may call CC (or its inclusion CXC \hookrightarrow X) a closed subspace. More abstractly, a subspace AA of a space XX is closed if the inclusion map AXA \hookrightarrow X is a closed map.

The collection of closed subsets of a space XX is closed under arbitrary intersections. If AXA \subseteq X, then the intersection of all closed subsets containing AA is the smallest closed subset that contains AA, called the topological closure of AA, and variously denoted Cl(A)Cl(A), Cl X(A)Cl_X(A), A¯\bar{A}, A¯\overline{A}, etc. It follows that ABA \subseteq B implies Cl(A)Cl(B)Cl(A) \subseteq Cl(B) and Cl(Cl(A))=Cl(A)Cl(Cl(A)) = Cl(A), so that ACl(A)A \mapsto Cl(A) forms a Moore closure operator on the power set P(X)P(X).

Since closed subsets are closed with respect to finite unions, we have Cl(AB)=Cl(A)Cl(B)Cl(A \cup B) = Cl(A) \cup Cl(B).

A topological closure operator is a Moore closure operator Cl:P(X)P(X)Cl: P(X) \to P(X) that preserves finite unions (Cl(0)=0Cl(0) = 0 and Cl(AB)=Cl(A)Cl(B)Cl(A \cup B) = Cl(A) \cup Cl(B)). It is easy to see that all such closure operators come from a topology whose closed sets are the fixed points of ClCl.

(There is a lot more to say, about convergence spaces, smooth spaces, schemes, etc.)

Definition

Definition

(closed subsets)

Let (X,τ)(X,\tau) be a topological space.

  1. A subset SXS \subset X is called a closed subset if its complement XSX \setminus S is an open subset:

    (SXis closed)AAAA(XSXis open). \left( S \subset X\,\, \text{is closed} \right) \phantom{AA} \Leftrightarrow \phantom{AA} \left( X\setminus S \, \subset X \,\, \text{is open} \right) \,.

    graphics grabbed from Vickers 89

  2. If a singleton subset {x}X\{x\} \subset X is closed, one says that xx is a closed point of XX.

  3. Given any subset SXS \subset X, then its topological closure Cl(S)Cl(S) is the smallest closed subset containing SS:

    Cl(S)CXclosedSC(C). Cl(S) \;\coloneqq\; \underset{ {C \subset X\, \text{closed} } \atop {S \subset C } }{\cap} \left( C \right) \,.
  4. A subset SXS \subset X such that Cl(S)=XCl(S) = X is called a dense subset of (X,τ)(X,\tau).

Properties

Basic properties

Lemma

(alternative characterization of topological cosures)

Let (X,τ)(X,\tau) be a topological space and let SXS \subset X be a subset of its underlying set. Then a point xXx \in X is contained in the topological closure Cl(S)Cl(S) (def. 1) precisely if every open neighbourhood U xXU_x \subset X of xx intersects SS:

(xCl(S))AAAA¬(UXSUXopen(xU)). \left( x \in Cl(S) \right) \phantom{AA} \Leftrightarrow \phantom{AA} \not\left( \underset{ {U \subset X \setminus S} \atop { U \subset X \, \text{open} } }{\exists} \left( x \in U \right) \right) \,.
Proof

Due to de Morgan duality we may rephrase the definition of the topological closure as follows:

Cl(S) SCCXclosed(C) =UXSUXopen(XU) =X(UXSUXopenU). \begin{aligned} Cl(S) & \coloneqq \underset{ {S \subset C } \atop { C \subset X\,\text{closed} } }{\cap} \left(C \right) \\ & = \underset{ { U \subset X \setminus S } \atop {U \subset X\, \text{open}} }{\cap} \left( X \setminus U \right) \\ & = X \setminus \left( \underset{ {U \subset X \setminus S} \atop { U \subset X\, \text{open} }}{\cup} U \right) \end{aligned} \,.
Proposition

(closure of a finite union is the union of the closures)

For II a finite set and {U iX} iI\{U_i \subset X\}_{i \in I} is a finite set of subsets of a topological space, then

Cl(iIU i)=iICl(U i). Cl(\underset{i \in I}{\cup}U_i) = \underset{i \in I}{\cup} Cl(U_i) \,.
Proof

By lemma 1 we use that a point is in the closure of a set precisely if every open neighbourhood of the point intersects the set.

Hence in one direction

iICl(U i)Cl(iIU i) \underset{i \in I}{\cup} Cl(U_i) \subset Cl(\underset{i \in I}{\cup}U_i)

because if every neighbourhood of a point intersects all the U iU_i, then every neighbourhood intersects their union.

The other direction

Cl(iIU i)iICl(U i) Cl(\underset{i \in I}{\cup}U_i) \subset \underset{i \in I}{\cup} Cl(U_i)

is equivalent by de Morgan duality to

XiICl(U i)XCl(iIU i) X \setminus \underset{i \in I}{\cup} Cl(U_i) \subset X \setminus Cl(\underset{i \in I}{\cup}U_i)

On left now we have the point for which there exists for each iIi \in I a neighbourhood U x,iU_{x,i} which does not intersect U iU_i. Since II is finite, the intersection iIU x,i\underset{i \in I}{\cap} U_{x,i} is still an open neighbourhood of xx, and such that it intersects none of the U iU_i, hence such that it does not intersect their union. This implis that the given point is contained in the set on the right.

In metric spaces

Proposition

Using classical logic then:

Let (X,d)(X,d) be a metric space, regarded as a topological space via its metric topology, and let VXV \subset X be a subset. Then the following are equivalent:

  1. VXV \subset X is a closed subspace.

  2. For every sequence x iVXx_i \in V \subset X with elements in VV, which converges as a sequence in XX it also converges in VV.

Proof

First assume that VXV \subset X is closed and that x iix x_i \overset{i \to \infty}{\longrightarrow} x_{\infty} for some x Xx_\infty \in X. We need to show that then x Vx_\infty \in V. Suppose it were not, then x X\Vx_\infty \in X\backslash V. Since by definition this complement X\VX \backslash V is an open subset, it follows that there exists a real number ϵ>0\epsilon \gt 0 such that the open ball around xx of radius ϵ\epsilon is still contained in the complement: B x (ϵ)X\VB^\circ_x(\epsilon) \subset X \backslash V. But since the sequence is assumed to converge in XX, this means that there exists N ϵN_\epsilon such that all x i>N ϵx_{i \gt N_{\epsilon}} are in B x (ϵ)B^\circ_x(\epsilon), hence in X\VX\backslash V. This contradicts the assumption that all x ix_i are in VV, and hence we have proved by contradiction that x Vx_\infty \in V.

Conversely, assume that for all sequences in VV that converge to some x Xx_\infty \in X then x VWx_\infty \in V \subset W. We need to show that then VV is closed, hence that X\VXX \backslash V \subset X is an open subset, hence that for every xX\Vx \in X \backslash V we may find a real number ϵ>0\epsilon \gt 0 such that the open ball B x (ϵ)B^\circ_x(\epsilon) around xx of radius ϵ\epsilon is still contained in X\VX \backslash V. Suppose on the contrary that such ϵ\epsilon did not exist. This would mean that for each kk \in \mathbb{N} with k1k \geq 1 then the intersection B x (1/k)VB^\circ_x(1/k) \cap V is non-empty. Hence then we could choose points x kB x (1/k)Vx_k \in B^\circ_x(1/k) \cap V in these intersections. These would form a sequence which clearly converges to
the original xx, and so by assumption we would conclude that xVx \in V, which violates the assumption that xX\Vx \in X \backslash V. Hence we proved by contradiction X\VX \backslash V is in fact open.

Relation to interior subspaces

Lemma

Let (X,τ)(X,\tau) be a topological space and let SXS \subset X be a subset. Then the topological interior Int(S)Int(S) of SS equals the complement of the topological closure Cl(X\S)Cl(X\backslash S) of the complement of SS:

Int(S)=X\Cl(X\S). Int(S) = X \backslash Cl\left( X \backslash S \right) \,.
Proof

By taking complements once more, the statement is equivalent to

X\Int(S)=Cl(X\S). X \backslash Int(S) = Cl( X \backslash S ) \,.

Now we compute:

X\Int(S) =X\(UopenUSU) =USX\U =CclosedCX\SC =Cl(X\S) \begin{aligned} X \backslash Int(S) & = X \backslash \left( \underset{{U \, open} \atop {U \subset S}}{\cup}U \right) \\ & = \underset{U \subset S}{\cap} X \backslash U \\ & = \underset{{C\, closed} \atop {C \supset X \backslash S}}{\cap} C \\ & = Cl(X \backslash S) \end{aligned}

Relation to compact subspaces

The relation of closed subspaces to compact subspaces is expressed by the following statements

Kuratowski’s closure-complement problem

This mildly amusing curiosity asks how many set-theoretic operations on a topological space XX are derivable from closure CC and complementation ¬\neg and applying finite composition. The answer is that at most 14 operations are so derivable (and there are examples showing this number is achievable). As the proofs below indicate, this bare fact has little to do with topology; it has more to do with general Moore closures and how they interact with complements (using classical logic).

Let P(X)P(X) denote the power set (ordered by inclusion) and MM the monoid of endofunctions P(X)P(X)P(X) \to P(X) with order defined pointwise. Then C 2=CC^2 = C and ¬ 2=1\neg^2 = 1 in MM, with CC order-preserving and ¬\neg order-reversing. Also

  • I¬C¬I \coloneqq \neg C \neg is the interior operation, with IIdI \leq Id.
Proposition

C¬C¬C \neg C \neg is idempotent.

Proof

I(C¬C)(C¬C)I (C \neg C) \leq (C \neg C), i.e., ¬C¬C¬CC¬C\neg C \neg C \neg C \leq C \neg C. Applying the order-preserving operation CC to both sides together with the fact that C 2=CC^2 = C, this gives

C¬C¬C¬CCC¬C=C¬C.C\neg C \neg C \neg C \leq C C \neg C = C \neg C.

Since ICCI C \leq C, we have also ¬C¬CC\neg C \neg C \leq C. Applying the order-reversing operation C¬CC \neg C to both sides, we obtain

C¬C=C¬CCC¬C(¬C¬C).C \neg C = C \neg C C \leq C \neg C (\neg C \neg C).

Combining the two displayed inequalities gives C¬C=C¬C¬C¬CC \neg C = C \neg C \neg C \neg C, and then multiplying this on the right by ¬\neg, the proposition follows.

Proposition

Let KK be the monoid presented by two generators C,¬C, \neg and subject to the relations C 2=CC^2 = C, ¬ 2=1\neg^2 = 1, and C¬C¬C¬C=C¬CC \neg C \neg C \neg C = C \neg C. Then KK, called the Kuratowski monoid, has at most 14 elements.

Proof

We may apply an obvious reduction algorithm on the set of words in two letters C,¬C, \neg, in which a word is reduced by replacing any substring CCC C by CC and any substring ¬¬\neg\neg by an empty substring, so that any word which cannot be further reduced must be alternating in C,¬C, \neg. This leads to a list of 14 words

1,¬,C,¬C,C¬,¬C¬,C¬C,1, \qquad \neg, \qquad C, \qquad \neg C, \qquad C \neg, \qquad \neg C \neg, \qquad C \neg C,
\,
¬C¬C,C¬C¬,¬C¬C¬,C¬C¬C,¬C¬C¬C,C¬C¬C¬,¬C¬C¬C¬\neg C \neg C, \qquad C \neg C \neg, \qquad \neg C \neg C \neg, \qquad C \neg C \neg C, \qquad \neg C \neg C \neg C, \qquad C \neg C \neg C \neg, \qquad \neg C \neg C \neg C \neg

with any further alternating words reducible by replacing a substring C¬C¬C¬CC \neg C \neg C \neg C by C¬CC \neg C. Thus each element in the monoid KK is represented by one of these 14 words.

These 14 words actually name distinct set-theoretic operations P(X)P(X)P(X) \to P(X) for a judicious choice of space XX; as a corollary, the Kuratowski monoid KK has exactly 14 elements. For instance (courtesy of Wikipedia), taking X=X = \mathbb{R} with its standard topology, the orbit of the element (0,1)(1,2){3}([4,5])(0, 1) \cup (1, 2) \cup \{3\} \cup ([4, 5] \cap \mathbb{Q}) under the monoid action consists of 14 distinct elements.

Remark

At most 7 operations are possible with interior and closure, corresponding to the covariant Kuratowski operations. Thus there is a 7-element submonoid K covKK_{cov} \hookrightarrow K. Spaces XX for which the topological action K covSet(P(X),P(X))K_{cov} \to Set(P(X), P(X)) is not injective are of some structural interest; for instance, the spaces for which CIC=ICC I C = I C are the extremally disconnected spaces, whereas spaces for which IC=CI C = C are those where the open sets are equivalence classes for some equivalence relation (partition spaces). Those for which I=CI = C are discrete spaces.

Remark

A more manifestly topological consideration is what happens when we throw joins (or meets) into the I,CI, C mix. Briefly, at most 13 subsets can be obtained by starting with a subset AP(X)A \in P(X) and generating new subsets by taking closures, interiors, and unions; the order structure of these 13 subsets coincides with the free cocompletion of the finite ordered monoid K covK_{cov} with respect to nonempty joins. Here we must use distributivity of CC over joins.

For some details on these remarks (and quite a bit more), see Gardner and Jackson, 2008 and Sherman 2004. An example of a non-topological Moore closure where the 14 operations are all distinct is given here.

Generalizations

Locales

In locale theory, every open UU in a locale XX defines a closed sublocale CU\mathsf{C} U which is given by the closed nucleus

j CU:VUV. j_{\mathsf{C} U}\colon V \mapsto U \cup V .

The idea is that CU\mathsf{C}U is the part of XX which does not involve UU (hence the notation CU\mathsf{C}U, or any other notation for a complement), and we may identify VV with UVU \cup V when we are looking only away from UU.

The sublocale CU\mathsf{C}U is literally a complement of UU in the lattice of sublocales of XX, i.e. UCU=U\cap \mathsf{C}U = \emptyset and UCU=XU\cup \mathsf{C}U = X as sublocales. Moreover, if XX is a (sober) topological space regarded as a locale, then the locale UU is also spatial, and so is CU\mathsf{C}U, corresponding exactly to the topological closed set XUX\setminus U. (The fixed points of j CUj_{\mathsf{C}U} can be identified with the open sets containing UU, which are bijectively related to the open subsets of XUX\setminus U.) Thus there is really only one notion of “closed subspace” whether we regard XX as a space or as a locale (at least as long as XX is sober).

Constructive mathematics

In constructive mathematics, however, there are many possible inequivalent definitions of a closed subspace, including:

  1. A subspace CXC\subset X is closed if it is the complement of an open subspace, i.e. if C=XUC = X\setminus U for some open subspace UU;
  2. A subspace CXC\subset X is closed if its complement XCX\setminus C is open;
  3. A subspace CXC\subset X is closed if it contains all its limit points, i.e. if for any xXx\in X such that UCU\cap C is inhabited for all neighborhoods UU of xx, we have xCx\in C.

Definition (1) coincides with definition (2) or (3) only if excluded middle holds, since under (2) or (3) every subspace of a discrete space is closed, while under (1) the only closed subspaces are those that are complements, and if every proposition is a negation then the law of double negation follows. On the other hand, Definition (2) is clearly too strong, because even closed intervals [a,b][a,b] in the real numbers can’t be proved constructively to satisfy it (though they do satisfy definitions (1) and (3)). Note also that Definition (1) always implies Definition (3), since if C=XUC = X\setminus U and every open neighborhood of xx intersects CC, then xUx\notin U and thus xCx\in C. Thus it is not unreasonable (see also below) to define:

  • CXC\subset X is strongly closed if it is the complement of an open subspace.
  • CXC\subset X is weakly closed if it contains all its limit points.

This constructive variety of notions of closed subspace gives rise to a corresponding variety of notions of Hausdorff space when applied to the diagonal subspace.

Note also that neither of the sensible constructive definitions behaves quite like closed subspaces do classically. In particular, neither of them is apparently stable under finite unions (though the too-strong Definition (2) is). The situation is better for locales; see below.

Locales in constructive mathematics

Of the “topological” definitions of closed subspace above, it is “strongly closed” (and the too-strong Definition (2)) that seem closest to the localic one. However, in the topological definitions we may not have X=U(XU)X = U \cup (X\setminus U) or X=(XC)CX = (X\setminus C) \cup C even as sets, whereas it remains true constructively that X=UCUX = U \cup \mathsf{C}U in the lattice of sublocales. In fact, we have the following:

Theorem

The following are equivalent:

  1. The law of excluded middle.
  2. Every closed sublocale of a spatial locale is spatial.
  3. Every closed sublocale of a discrete locale is spatial.
Proof

We remarked above that (1)\Rightarrow(2), and of course (2)\Rightarrow(3). So assume (3). Every spatial sublocale is a union of its points, and in a discrete space points are open; thus if closed sublocales are spatial, they are also open. Since X=UCUX = U \cup \mathsf{C}U is constructively true, it follows that every open set is complemented in the open-set lattice of any discrete locale, which is to say that all powersets are Boolean algebras, i.e. excluded middle holds.

On the other hand, there is a localic notion of weakly closed sublocale that is closely related to topologically weakly closed subspaces (so that the above notion of “closed sublocale” — the formal complement of an open sublocale — could also be called strongly closed). It is the specialization of the notion of fiberwise closed sublocale? to locale maps X1X\to 1 into the terminal object.

Definition

A sublocale CXC\subseteq X is weakly closed if it is not strongly dense in any larger sublocale of XX. That is, if whenever DD is a sublocale of XX such that CDC\subseteq D by a strongly dense inclusion, then C=DC=D.

Since strong and weak denseness coincide classically, so do strong and weak closedness. And as we expect, strong closedness implies weak closedness, since strong density implies weak density. Moreover, both of them are better-behaved than the corresponding topological notions. For instance, strongly and weakly closed sublocales are both stable under finite unions (in the lattice of sublocales), even constructively.

Both strongly and weakly closed sublocales are “correct” notions; they are simply different. Some classical theorems about closed sublocales are constructively about strongly closed ones, while others (such as the theorem that any subgroup of a localic group is closed) are about weakly closed ones.

However, as we saw above, not every strongly closed sublocale (hence not every weakly closed sublocale either) can be spatial. But it is shown at strongly dense sublocale that a subspace of a space is localically strongly dense if and only if it is topologically strongly dense. This leads us to guess:

Conjecture

A subspace CXC\subseteq X of a (sober) topological space XX is topologically weakly closed if and only if it is the spatial coreflection of a weakly closed sublocale.

In one direction this is easy: suppose CC is topologically weakly closed, and let DD be its localic weak closure. This is, by definition, the largest sublocale of XX in which CC is localically strongly dense. Now let EE be the spatial coreflection of DD; since CC is spatial we have CEC\subseteq E, and since CC is localically strongly dense in DD, it is also so in EE, and hence topologically strongly dense. But CC is also topologically weakly closed, hence C=EC=E and is the spatial coreflection of the weakly closed sublocale DD.

The other direction is harder.

References

  • C. Kuratowski, Sur l’opération A¯\bar{A} de l’analysis situs , Fund. Math. III (1922) pp.192-195. (pdf)

Weakly closed sublocales are discussed in

  • Sketches of an Elephant, C1.1 and C1.2

  • Peter Johnstone, A constructive ‘closed subgroup theorem’ for localic groups and groupoids, Cahiers de Topologie et Géométrie Différentielle Catégoriques (1989), Volume: 30, Issue: 1, page 3-23 link

  • M. Jibladze and Peter Johnstone, The frame of fibrewise closed nuclei, Cahiers de Topologie et Géométrie Différentielle Catégoriques (1991), Volume: 32, Issue: 2, page 99-112, link

  • Peter Johnstone, Fiberwise separation axioms for locales

Revised on May 23, 2017 08:15:12 by Urs Schreiber (131.220.184.222)