UDC 513.83+519.54

MATHEMATICS

V. V. FEDORCHUK

UNIFORM SPACES AND PERFECT IRREDUCIBLE MAPPINGS OF TOPOLOGICAL SPACES

(Presented by Academician P. S. Aleksandrov, 15 XII 1969)

This note introduces the concept of a \(\theta\)-uniform space. This concept generalizes the notion of a uniform space compatible with a given topological space. \(\theta\)-uniformities exist on every Hausdorff space. Just as every uniformity naturally generates a proximity, every \(\theta\)-uniformity generates a \(\theta\)-proximity.*

As in the case of \(\theta\)-proximities, there is a one-to-one correspondence between \(\theta\)-uniformities on a given topological space and uniformities on its completely regular preimages with respect to \(\theta\)-perfect irreducible mappings. In this situation ordinary uniformities are distinguished among all \(\theta\)-uniformities as projective objects with respect to the class of uniform regular \(\theta\)-mappings.

§ 1. We shall say that a system \(\nu=\{V\}\) of canonically open subsets of a topological space \(X\) is a \(\theta\)-covering of the space \(X\) of locally finite type if, for every point \(x\in X\), there exists a finite collection \(V_1,\ldots,V_n\) of elements of the system \(\nu\) such that

\[ x\in \left\langle \bigcup_{i=1}^{n} [V_i]\right\rangle . \]

A large stock of \(\theta\)-coverings of locally finite type is provided by

Lemma 1. Let \(f:Y\to X\) be a \(\theta\)-perfect irreducible mapping and let \(\nu=\{V\}\) be a covering of the space \(Y\) by canonically open sets. Then the system \(f^{\#}\nu=\{f^{\#}V\mid V\in\nu\}\) will be a \(\theta\)-covering of locally finite type of the space \(X\).

A family \(\mathfrak{B}=\{\nu\}\) of \(\theta\)-coverings of locally finite type of a topological space \(X\) is called a \(\theta\)-uniformity if the following axioms are satisfied:

\(\mathrm{I}_{\mathrm{U}}\). If a \(\theta\)-covering \(\nu\in\mathfrak{B}\) is inscribed in a \(\theta\)-covering \(w\) of locally finite type, then \(w\in\mathfrak{B}\).

\(\mathrm{II}_{\mathrm{U}}\). For any two \(\theta\)-coverings \(u,\nu\in\mathfrak{B}\) there exists a \(\theta\)-covering \(w\in\mathfrak{B}\) which is star-inscribed both in the \(\theta\)-covering \(u\) and in the \(\theta\)-covering \(\nu\).

\(\mathrm{III}_{\mathrm{U}}\). If \(x,y\) are distinct points of the space \(X\), then there exist neighborhoods \(G\) and \(H\) of these points and a \(\theta\)-covering \(\nu\in\mathfrak{B}\) such that
\[ H\cap \operatorname{st}_{\nu}G=\varnothing . \]

\(\mathrm{IV}_{\mathrm{U}}\). For every point \(x\in X\) and for each of its canonically open neighborhoods \(G\) there exists a neighborhood \(H\) of the point \(x\) and a \(\theta\)-covering \(\nu\in\mathfrak{B}\) such that \(\operatorname{st}_{\nu}H\subset G\). Thus, if in the axioms of \(\theta\)-uniformity the \(\theta\)-coverings of locally finite type are replaced by coverings, one obtains one of the variants of the axioms of a separated uniformity compatible with the given topology.

* \(\theta\)-proximities are studied in detail in work \((^{1})\), whose terminology and notation we use here as well.

A topological space \(X\) endowed with a \(\theta\)-uniformity \(\mathfrak{B}\) is called a \(\theta\)-uniform space \((X,\mathfrak{B})\). A fundamental system or base of a \(\theta\)-uniformity is any set \(\mathfrak{B}_1\subset \mathfrak{B}\) having the property that into every \(\theta\)-cover \(v\in\mathfrak{B}\) one can inscribe a \(\theta\)-cover \(v_1\in\mathfrak{B}_1\). From a fundamental system the \(\theta\)-uniformity is uniquely recovered by virtue of axiom I\(_U\).

Every separated uniformity is a \(\theta\)-uniformity. On an extremally disconnected space every \(\theta\)-uniformity is a uniformity, since every locally finite \(\theta\)-cover of an extremally disconnected space is a cover. An example of a \(\theta\)-uniformity on an arbitrary Hausdorff space \(X\) is the \(\theta\)-uniformity whose base is the system of all finite \(\theta\)-covers by canonically open sets.

Every \(\theta\)-uniformity \(\mathfrak{B}\) on a space \(X\) generates a \(\theta\)-proximity:
\(A\theta B \Longleftrightarrow\) there exist neighborhoods \(C\) and \(D\) of the sets \(A\) and \(B\), respectively, and a \(\theta\)-cover \(v\in\mathfrak{B}\) such that \(C\cap \operatorname{st}_v D=\varnothing\). We shall denote this \(\theta\)-proximity by the symbol \(\theta_{\mathfrak{B}}\).

The most general example of a \(\theta\)-uniformity is given by:

Theorem 1. Let \(f:Y\to X\) be a \(\theta\)-perfect irreducible mapping of a completely regular space \(Y\) onto a space \(X\). Let a uniformity \(\mathfrak{B}\) be given on the space \(Y\). Then the family
\[ f^{\#}\mathfrak{B}=\{\,f^{\#}v\mid v\in\mathfrak{B},\ v\text{ consists of canonically open sets}\,\} \]
is a \(\theta\)-uniformity on the space \(X\). Moreover,
\[ \theta_{f^{\#}\mathfrak{B}}=f\theta_{\mathfrak{B}} \,^{*}. \]

The following theorem shows that there exists only one uniformity generating a given \(\theta\)-uniformity.

Theorem 2. Let \(f_i:Y_i\to X\) be \(\theta\)-perfect irreducible mappings of completely regular spaces onto the space \(X\), \(i=1,2\). Let uniformities \(\mathfrak{B}_i\) be given on the spaces \(Y_i\) such that
\[ f_1^{\#}\mathfrak{B}_1=f_2^{\#}\mathfrak{B}_2. \]
Then there exists a uniformly continuous in both directions homeomorphism \(h:Y_1\to Y_2\) such that \(f_1=f_2h\).

Theorem 3 asserts that every \(\theta\)-uniformity is generated by some uniformity, unique by Theorem 2.

Theorem 3. Let \((X,\mathfrak{B})\) be a \(\theta\)-uniform space. There exists a uniform space \((X_{\mathfrak{B}},\widetilde{\mathfrak{B}})\) and a \(\theta\)-perfect irreducible mapping \(\pi_{\mathfrak{B}}:X_{\mathfrak{B}}\to X\) such that
\[ \mathfrak{B}=\pi_{\mathfrak{B}}^{\#}\widetilde{\mathfrak{B}}. \]

§ 2. A mapping \(f:(X,\mathfrak{B})\to(Y,\mathfrak{M})\) is called \(\theta\)-uniformly continuous if: 1) for every \(\theta\)-cover \(w\in\mathfrak{M}\) there exists a \(\theta\)-cover \(v\in\mathfrak{B}\) such that the cover \(\{f[V]\mid V\in v\}\) is inscribed in the cover \(\{W\mid W\in\mathfrak{M}\}\); 2) the family \(\{\langle[f^{-1}W]\rangle\mid W\in w\}\) is a uniform \(\theta\)-cover of the space \((X,\mathfrak{B})\).

Lemma 2. If \(f:(X,\mathfrak{B})\to(Y,\mathfrak{M})\) is a \(\theta\)-uniformly continuous mapping, then the mapping
\[ f:(X,\theta_{\mathfrak{B}})\to(Y,\theta_{\mathfrak{M}}) \]
is \(\theta\)-proximally continuous.

The following theorem is analogous to Theorem 4 of [1] and is proved with its aid.

Theorem 4. Let \(f:(X,\mathfrak{B})\to(Y,\mathfrak{M})\) be a \(\theta\)-uniformly continuous mapping onto a regular space \(Y\). Then there exists a uniformly continuous mapping
\[ \widetilde f:(X_{\mathfrak{B}},\widetilde{\mathfrak{B}})\to(Y_{\mathfrak{M}},\widetilde{\mathfrak{M}}) \]
such that the diagram is commutative
\[ \begin{array}{ccc} X_{\mathfrak{B}} & \xrightarrow{\ \widetilde f\ } & Y_{\mathfrak{M}}\\ \downarrow{\pi_{\mathfrak{B}}} & & \downarrow{\pi_{\mathfrak{M}}}\\ X & \xrightarrow{\ f\ } & Y \end{array} \]

* By \(\delta_{\mathfrak{B}}\) is denoted the proximity generated on the space \(Y\) by the uniformity \(\mathfrak{B}\), and by \(f\theta_{\mathfrak{B}}\) the \(\theta\)-proximity generated on \(X\) by this proximity and the mapping \(f:Y\to X\) (see Theorem 1 of [1]).

Remark. The requirement of regularity of the space \(Y\) in this theorem can be weakened only to the requirement of continuity of the mapping \(f\), since there exists an example showing that Theorem 4 of \((^{1})\), which follows from the present theorem, ceases to be true if the regularity of the space \(Y\) is omitted. The regularity of the space \(Y\) can be omitted for closed and irreducible mappings.

Theorem 5. Let \(f:(X,\mathfrak V)\to (Y,\mathfrak W)\) be a \(\theta\)-uniformly continuous mapping of the space \(X\) onto the space \(Y\). Suppose, moreover, that the mapping \(f\) is closed\(^*\) and irreducible. Then there exists a uniformly continuous mapping
\[ \widetilde f:(X_{\mathfrak V},\widetilde{\mathfrak V})\to (Y_{\mathfrak W},\widetilde{\mathfrak W}) \]
such that
\[ \pi_{\mathfrak W}\widetilde f=f\pi_{\mathfrak V}. \]
If, in addition, the mapping \(f\) is bicompact, then the mapping \(\widetilde f\) is perfect and irreducible.

A mapping \(f:(X,\mathfrak V)\to (Y,\mathfrak W)\) is called \(\theta\)-uniformly regular if it is \(\theta\)-uniformly continuous and, for every uniform \(\theta\)-covering \(v\in\mathfrak V\), there exists a uniform \(\theta\)-covering \(w\in\mathfrak W\) that is inscribed in the system \(f^\#v\).

We shall call a \(\theta\)-perfect irreducible mapping \(f:(X,\mathfrak V)\to (Y,\mathfrak W)\) a uniform \(\theta\)-mapping if the mapping \(f\) is \(\theta\)-uniformly continuous. Finally, a \(\theta\)-uniformly regular uniform \(\theta\)-mapping will be called a uniform regular \(\theta\)-mapping.

Theorem 6. The uniform space \((X_{\mathfrak V},\widetilde{\mathfrak V})\) is a projective object in the class of all \(\theta\)-uniform spaces \((Y,\mathfrak W)\) mapped onto the space \((X,\mathfrak V)\) by uniform regular \(\theta\)-mappings. Moreover, if \(f:(Y,\mathfrak W)\to (X,\mathfrak V)\) is a uniform regular \(\theta\)-mapping, then in the commutative diagram
\[ \begin{array}{ccc} Y_{\mathfrak W} & \xrightarrow{\ \widetilde f\ } & X_{\mathfrak V}\\ \downarrow{\pi_{\mathfrak W}} & & \downarrow{\pi_{\mathfrak V}}\\ Y & \xrightarrow{\ f\ } & X \end{array} \]
the mapping \(\widetilde f\) is a uniformly continuous homeomorphism in both directions.

From this theorem follows the theorem on the \(\theta\)-absolute (see \((^{1})\)), since every regular \(\theta\)-mapping is a uniform regular \(\theta\)-mapping with respect to the precompact \(\theta\)-uniformities.

Faculty of Mechanics and Mathematics
M. V. Lomonosov Moscow State University

Received
10 XII 1969

REFERENCES

1. V. Fedorchuk, Matem. sborn., 76 (118), 4, 513 (1968).
2. V. Fedorchuk, DAN, 180, No. 3 (1968).

\(^*\) Here the closedness of the mapping \(f\) does not presuppose its continuity.