Mathematics

O. V. LOKUTSIEVSKII

THE SPACE OF RELATIVE TOPOLOGY*

(Presented by Academician P. S. Aleksandrov on 20 III 1964)

In (⁴) and independently, but somewhat later, in (⁵)**, a topologically invariant concept was defined that generalizes the concept of a space of relative metric of Mazurkiewicz–Urysohn (see (¹), p. 559, and (²), p. 171). Below we give the definition of this concept and state a number of assertions established in studying it.

1. Let \(X_\rho\) be a bounded metric space***. C. Mazurkiewicz calls the relative distance between two of its points the lower bound of the diameters of connected sets containing these points, if such sets exist, and the diameter of \(X_\rho\) itself otherwise. The relative distance turns the set \(X\) into a new metric space \(X_{\tilde{\rho}}\)—the space of the relative metric \(X_\rho\).

It is easy to give the concept of a space of relative metric a topologically invariant character. Namely, let \(X_\tau\) be a topological space, \(g=\{G_\alpha\}\) its base. It turns out that the system \(\tilde g\), consisting of the components of the elements of this base, generates on the set \(X\) some (generally speaking, new) topology and that this topology does not depend on the choice of the base \(g\) of the original space \(X_\tau\). The topological space thus obtained will henceforth be denoted by \(X_{\tilde{\tau}}\) and will be called the space of relative topology of the space \(X_\tau\). Its connection with the space of relative metric is described by

Theorem 1. If \(X_\rho\) is a bounded metric space, then the identity mapping
\[ \xi: X_{\tilde{\tau}(\rho)} \to X_{\tau(\tilde{\rho})} \]
is a homeomorphism.

1. Let us turn to the study of the space \(X_{\tilde{\tau}}\). It is easily verified that the identity mapping \(\xi: X_{\tilde{\tau}}\to X_\tau\) is a homeomorphism if and only if \(X_\tau\) is locally connected. (This mapping is, obviously, always continuous.)

Theorem 2. If \(X_\tau\) is Hausdorff (regular, completely regular), then \(X_{\tilde{\tau}}\) is also Hausdorff (regular, completely regular)****.

Theorem 3*****. If \(X_\tau\) is Hausdorff, then \(X_{\tilde{\tau}}\) is bicompact if and only if \(X_\tau\) is a locally connected bicompactum.

* Reported (except for item 3) at the All-Union Topological Conference on 30 IX 1963.

** I learned about paper (⁴) from Z. Frolík after my report at the topological conference (and, consequently, already after the publication of (⁵)). A substantial part of the overlap of results is contained in Theorems 1 and 4.

*** The letters \(X\) and \(Y\) denote sets. The assignment on any one of them of a metric (or topological) structure is indicated by the lower index \(\rho\) (or \(\tau\)). Different metric (topological) structures of one and the same set are distinguished by means of symbols attached to these indices. (For example: \(X_\rho\), \(X_{\tilde{\rho}}\), or \(Y_{\tau_1}\), \(Y_{\tau_2}\), etc.). If \(X_\rho\) is a metric space, then the topological space generated by it is denoted by \(X_{\tau(\rho)}\).

**** It is unknown to me whether normality of \(X_\tau\) implies normality of \(X_{\tilde{\tau}}\). Metrizability of \(X_{\tilde{\tau}}\) follows from metrizability of \(X_\tau\) by virtue of Theorem 1.

***** For the space of relative metric of the continuum an analogous assertion was proved by P. S. Urysohn (see (¹), p. 560).

Very useful is

Theorem 4. If the mapping \(f: X_{\tau_1}\to Y_{\tau_2}\) is continuous, then the mapping \(f:X_{\widetilde{\tau}_1}\to Y_{\widetilde{\tau}_2}\) is also continuous.

Corollary 1. If \(X_\tau\subseteq Y_\tau\), then the identity mapping
\(\xi:X_{\widetilde{\tau}}\to Y_{\widetilde{\tau}}\) is continuous.

3. Definition 1. A subspace \(X_\tau\) of a space \(Y_\tau\) is called properly situated in \(Y_\tau\) if the identity mapping \(\xi:X_{\widetilde{\tau}}\to Y_{\widetilde{\tau}}\) is a homeomorphism.

For what follows the following are convenient.

Notation. Let \(x\in A\subseteq X_\tau\). By \(c_x^\tau A\) (respectively \(k_x^\tau A\)) we denote the component (respectively, quasicomponent) of the set \(A\) containing the point \(x\).

Theorem 5. A subspace \(X_\tau\) of a space \(Y_\tau\) is properly situated in \(Y_\tau\) if and only if, for every point \(x\in X_\tau\) and every neighborhood \(U\subseteq Y_\tau\) of it, there exists a smaller neighborhood \(V\subseteq Y_\tau\) of it such that
\[ X_\tau\cap (c_x^\tau V)\subseteq c_x^\tau(X_\tau\cap U). \]

Corollary 2. The following subsets of a topological space are properly situated in it: a) open subsets, b) locally connected subsets, c) components of open sets.

Theorem 6. A completely regular space \(X_\tau\) is properly situated in at least one of its bicompact extensions* if and only if, for every point \(x\in X_\tau\) and every neighborhood \(U\) of it, there exists a smaller neighborhood \(V\) of it such that
\[ k_x^\tau V\subseteq c_x^\tau U. \]

Let us note that if a completely regular space is properly situated in the smaller of two of its bicompact extensions, then it is properly situated also in the larger of them. In this connection the following is of interest.

Theorem 7. Among all those bicompact extensions of a completely regular space \(X_\tau\) in which it is properly situated, there exists a minimal one if and only if \(X_\tau\) is locally bicompact. In this case \(X_\tau\) is properly situated in all of its bicompact extensions.

4. In the study of properties of local connectedness and local bicompactness, the following plays an essential role.

Condition \((\chi)\). A space \(X_\tau\) is said, by definition, to satisfy condition \((\chi)\) at a point \(x\in X_\tau\) if there exists a neighborhood \(U\) of this point such that the set \(c_x^\tau U\) is locally connected and locally bicompact.

Theorem 8. If a space \(X_\tau\) satisfies condition \((\chi)\) at every point, then \(X_{\widetilde{\tau}}\) is locally connected**.

The converse assertion does not hold.

In Theorem 9 and Corollary 3 the space \(X_\tau\) is assumed to be Hausdorff.

Theorem 9. The space \(X_{\widetilde{\tau}}\) is locally bicompact if and only if \(X_\tau\) satisfies condition \((\chi)\) at every point.

Corollary 3. If \(X_{\widetilde{\tau}}\) is locally bicompact, then it is locally connected.

5. Let \(X_\tau\) be bicompact. As was already said, the identity mapping
\(\xi:X_{\widetilde{\tau}}\to X_\tau\) is continuous. The existence of such bicompact extensions of the space \(X_{\widetilde{\tau}}\) (we shall call them branched) to which this mapping can be continuously extended is easily proved. It is also clear that if the smaller of two bicompact extensions

* Bicompact extensions are everywhere assumed to be Hausdorff.

** In fact, this assertion follows only from the first part of condition \((\chi)\), namely from the local connectedness of \(U\) (without the assumption that this set is locally bicompact).

extensions of the space \(X_{\widetilde{\tau}}\) is branched, then the larger one is also branched. Therefore the following is legitimate.

Definition 2. A bicompactum \(X_\tau\) is called branching if, among all branched bicompact extensions of the space \(X_{\widetilde{\tau}}\), there exists a minimal one.

Theorem 10. If the bicompactum \(X_\tau\) satisfies condition \((\chi)\) at every point, then it is branching.*

Simple examples show that condition \((\chi)\) is not necessary in order that the bicompactum \(X_\tau\) be branching. To formulate a necessary and sufficient criterion we introduce the following

Condition (K). The space \(X_\tau\) satisfies, by definition, condition (K) at a point \(x \in X_\tau\) if for every neighborhood \(U\) of this point there exists a smaller neighborhood \(V\) of it such that the set \(c_x^\tau V\) is locally connected at all points** belonging to the intersection
\((c_x^\tau V)\cap\{c_x^\tau U\setminus\langle c_x^\tau U\rangle\}\).

It is clear that condition (K) follows from condition \((\chi)\).

Theorem \(10^*\). A bicompactum \(X_\tau\) is branching if and only if it satisfies condition (K) at every point of local non-connectedness.

1. If \(X_\tau\) is a completely regular space, then the study of the space \(X_{\widetilde{\tau}}\) can be approached from the point of view of uniform structures***.

Suppose that the set \(X\) is endowed with a uniform structure \(\Sigma\)**** (see (³), p. 562). The uniform and topological spaces generated by this structure will be denoted respectively by \(X_\Sigma\) and \(X_{\tau(\Sigma)}\). (As is known from (³), \(X_{\tau(\Sigma)}\) is completely regular.)

To each covering \(\sigma\in\Sigma\) we assign the covering \(\widetilde{\sigma}\), consisting of the components* of the elements of \(\sigma\), and consider the system \(\widetilde{\Sigma}\) of all such coverings, into each of which a covering of the form \(\widetilde{\sigma}\) (where \(\sigma\in\Sigma\)) can be inscribed.

Theorem 11. The system \(\widetilde{\Sigma}\) forms a separated uniform structure.

Thus a new uniform space \(X_{\widetilde{\Sigma}}\) arises. The connection of the completely regular topological space \(X_{\tau(\widetilde{\Sigma})}\) generated by it with the space of relative topology is established by

Theorem 12. The identity mapping
\(\xi: X_{\tau(\widetilde{\Sigma})}\to X_{\widetilde{\tau(\Sigma)}}\) is a homeomorphism*.*

Corollary 4. If the mapping
\(f: X_{\tau(\Sigma_1)}\to Y_{\tau(\Sigma_2)}\) is continuous, then the mapping
\(f: X_{\tau(\widetilde{\Sigma}_1)}\to Y_{\tau(\widetilde{\Sigma}_2)}\) is also continuous.

Corollary 5. If the uniform structures \(\Sigma_1\) and \(\Sigma_2\) on the set \(X\) generate on it one and the same topology, then \(\widetilde{\Sigma}_1\) and \(\widetilde{\Sigma}_2\) also generate on \(X\) one and the same topology.

* By virtue of Theorem 9, this assertion follows from the following general fact: a completely regular space \(Y_\tau\) is locally bicompact if and only if, under any continuous mapping of it into any bicompactum, there exists a minimal one among all those bicompact extensions of \(Y_\tau\) to which this mapping can be continuously extended.

** A space \(Y_\tau\) is called locally connected at a point \(y\in Y_\tau\) if, for every neighborhood \(U\) of this point, the inclusion \(y\in\langle c_y^\tau U\rangle\) holds (see (²), p. 171).

*** Yu. M. Smirnov drew my attention to the possibility of such an approach. I take this opportunity to note that conversations with him were very useful to me in my work.

**** Only separated uniform structures are considered.

***** In the topology of the space \(X_{\tau(\Sigma)}\).

** Since every completely regular space is generated by some uniformity (see (³)), this also implies the part of the assertion of Theorem 2 that concerns completely regular spaces.

An analogue of Theorem 4 is

Theorem 13. If the mapping \(f: X_{\Sigma_1}\to Y_{\Sigma_2}\) is uniformly continuous, then the mapping \(f: X_{\widetilde{\Sigma}_1}\to Y_{\widetilde{\Sigma}_2}\) is also uniformly continuous.

7. Consider the \(\delta\)-space \(X_\delta\) (see (3)) and some uniform structure \(\Sigma\) of it. Then the \(\delta\)-space \(X_{\delta(\widetilde{\Sigma})}\) is uniquely determined (by analogy with \(X_{\tau(\widetilde{\Sigma})}\)). In particular, \(X_{\delta(\widetilde{\Sigma}^0)}\) is determined, where \(\Sigma^0\) is a precompact structure of the space \(X_\delta\). If \(X_\delta\) has a maximal structure \(\Sigma^\infty\), then \(X_{\delta(\widetilde{\Sigma}^\infty)}\) is also determined. In what follows we shall denote the spaces \(X_{\delta(\widetilde{\Sigma}^0)}\) and \(X_{\delta(\widetilde{\Sigma}^\infty)}\) respectively by \(X_{\delta^0}\) and \(X_{\delta^\infty}\).

A natural question arises: if \(\Sigma_1\) and \(\Sigma_2\) are uniform structures of one and the same \(\delta\)-space \(X_\delta\), do \(\widetilde{\Sigma}_1\) and \(\widetilde{\Sigma}_2\) generate the same closeness on the set \(X\)? In other words, is an analogue of Corollary 5 (or of the more general Corollary 4) true for \(\delta\)-spaces?

It turns out that it is not; namely: there exists a metrizable space of countable weight \(X_\rho\) such that the structures \(\widetilde{\Sigma}^0\) and \(\widetilde{\Sigma}^\infty\) generate different closenesses on the set \(X\).

Nevertheless, from Theorem 12 and Corollary 5 it follows that

Theorem 14. If \(\Sigma_1\) and \(\Sigma_2\) are uniform structures of one and the same \(\delta\)-space \(X_\delta\), then \(\widetilde{\Sigma}_1\) and \(\widetilde{\Sigma}_2\) generate on \(X\) one and the same topology, coinciding with the topology of each of the spaces \(X_{\tau(\widetilde{\Sigma}_1)}\) and \(X_{\tau(\widetilde{\Sigma}_2)}\).

The following assertions are also true:

Theorem 15. If \(f: X_{\delta_1}\to Y_{\delta_2}\) is a \(\delta\)-mapping, and \(\Sigma_1\) is an arbitrary uniform structure of \(X_{\delta_1}\), then \(f: X_{\delta(\widetilde{\Sigma}_1)}\to Y_{\delta_2^0}\) is also a \(\delta\)-mapping.

Corollary 6. If \(f: X_{\delta_1}\rightleftarrows Y_{\delta_2}\) is a \(\delta\)-homeomorphism, then \(f: X_{\delta_1^0}\rightleftarrows Y_{\delta_2^0}\) is also a \(\delta\)-homeomorphism.

For the spaces \(X_{\delta_1^\infty}\) and \(Y_{\delta_2^\infty}\) (if they exist) analogues of Theorem 15* and Corollary 6 hold.

Theorem 16. If \(X_\rho\) is a bounded metrizable space, then the closeness generated on \(X\) by the metric of the space \(X_\rho\) coincides with the closeness of the space \(X_{\delta(\widetilde{\Sigma}^{\infty})}\), where \(\Sigma^\infty\) is the metrizable** structure of \(X_\rho\).

CITED LITERATURE

1. P. S. Urysohn, Works on topology and other fields of mathematics, 1951.
2. F. Hausdorff, Set Theory, 1937.
3. Yu. M. Smirnov, Mat. sbornik, 31 (73), No. 3 (1952).
4. Z. Frolik, Czechoslovak Mathematical Journal, 11 (86), No. 3 (1961).
5. O. V. Lokutsievskii, DAN, 151, No. 4 (1963).

* The \(\delta\)-mapping turns out to be \(f: X_{\delta_1^\infty}\to Y_{\delta(\widetilde{\Sigma}_2)}\), where \(\Sigma_2\) is an arbitrary uniform structure of \(Y_{\delta_2}\).

** That is, maximal.