nLab
closed subspace

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


Contents

Definitions

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 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.)

Properties

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.
  4. Finally, we can also consider closed sublocales of the locale corresponding to XX.

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.

Note also that definition (2) is the classical contrapositive of definition (3): XCX\setminus C is open if for any xCx\notin C there exists a neighborhood UU of xx such that UC=U\cap C = \emptyset. This makes it seem unlikely that they are constructively equivalent, but I do not have a specific example.

Of the first three “topological” definitions, the closest to the localic one is definition (1), since both are defined as a “complement” of some open subspace. However, in definition (1) we may not have X=U(XU)X = U \cup (X\setminus U) 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.

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.

References

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

Revised on December 2, 2016 06:16:54 by Todd Trimble (67.81.95.215)