CC BY
3BULLETIN OF OSH STATE UNIVERSITY MATHEMATICS. PHYSICS. TECHNOLOGY. 2024, № 1(4)
УДК 515.122
DOI: https://doi.org/10.52754/16948645 2024 1(4) 46
ON COMPACTNESS TYPE EXTENSIONS OF TOPOLOGYCAL AND UNIFORM
SPACES
Chamashev Marat Kakarovich, Associate Professor
marat2771@mail.ru Osh State University, Osh, Kyrgyzstan Namazova Guliza Omurbekovna guliza_n@mail. ru Institute of Mathematics NAS KR, Bishkek, Kyrgyzstan
Abstract. In this article extensions of real-complete Tychonoff and uniform spaces are considered, as well as locally compact paracompact and locally compact Lindeloff extensions of Tychonoff and uniform spaces.
Keywords: Uniform real complete extension, locally compact paracompact extension, locally compact Lindeloff extension.
О РАСШИРЕНИЯХ ТИПА КОМПАКТНОСТИ ТОПОЛОГИЧЕСКИХ И РАВНОМЕРНЫХ ПРОСТРАНСТВ
ЧамашевМарат Какарович, к.ф.-м.н., доцент
marat2771@mail. ru Ошский государственного университет
Ош, Кыргызстан Намазова Гулиза Омурбековна, к.ф.-м.н., доцент
guliza_n@,mail. ru Институт математики НАН КР Бишкек, Кыргызстан
Абстракт. В этой статье рассматриваются расширения вещественно полных тихоновских и равномерных пространств, а также локально компактно паракомпактные и локально компактно линделёфовые расширения тихоновских и равномерных пространств.
Ключевые слова: Равномерно вещественно полное расширение, локально компактный паракомпакт, локально компактное линделёфово расширение.
ТОПОЛОГИЯЛЫК ЖАНА БИР КАЛЫПТУУ МЕЙКИНДИКТЕРДИН КОМПАКТУУЛУК ТYРYНYН КЕНЕЙТYYСY
Чамашев Марат Какарович, ф.-м.и.к., доцент
marat2771@mail. ru Ош мамлекеттик университети Ош, Кыргызстан Намазова Гулиза Омурбековна, ф.-м.и.к., доцент
guliza_n@mail. ru КР УИА Математика институту Бишкек, Кыргызстан
Абстракт. Бул макалада чыныгы толук тихоновдук жана бир калыптагы мейкиндиктерди кецейтYYлвр, ошондой эле тихоновдук жана бир калыптагы мейкиндиктердин жергиликтYY компакттуу паракомпакттуу жана жергиликтYY компакттуу линделвфтук кецейтYYлвр каралат.
ypyHmmyy co3dop. Eup кaммnmaгм Hbmbtebi mo^yK Ke^umyy, ^epгuмuкmYY KOMnaKmmyy napaKOMnaKm, weptumKmYY KOMnaKmmyy nmdenefymYK Ke^umYY.
The maximal real complete Tychonoff spaces extensions called the Hewitt extension. The real complete spaces introduced by Edwin Hewitt [1]. Uniform analogues analysis of other important classes topological spaces and formation all extensions of such Tychonoff spaces classes considered ([2],[3]).
Real complete extensions are considered in [4].
Let CU(X)- the set of all uniformly continuous functions /: (X,//) — ( R.¿r) and ER-natural uniformity of the number line R.
Definition 1. A uniform space (X, U) is called a uniformly functional space, and the uniformity U is functional if the uniformity U is generated by some family of functions CU(X), i.e. U is generated by a family of coverings of the form (/ _ : /e Q, (X) , a
Proposition 1. For every uniformity of U on X existence, uniformity of UF on X such that UF is the maximum functional uniformity contained in the uniformity of U.
Definition 2. A uniform space (X, U) is called uniformly real complete if it is uniformly functionally and complete.
Theorem 1. Let (X, U) be a uniformly function space. Then its completion (X, U) is uniformly real complete, and its topological space (X, ) will be real complete spaces.
Let (X, U) be an arbitrary uniform space. Then, by Proposition 1, there exists maximal functional uniformity i/F contained in U. By Theorem 1, the completions (X, U) of the uniform space (X, U) are uniformly real complete, and its topological space (X, U)is a real complete space. We denote it by VyX and call it the Hewitt extension of the uniform space (X, U).
If U is the maximal uniformity of the Tychonoff space X, then VyX coincides with the classical Hewitt extension IX of the Tychonoff space X.
Let X be an arbitrary real complete space. By C (X) we denote the set of all continuous functions f: X ^ R, which generates the maximum functional uniformity of UF. We show that the uniformity of UF is complete. By the external characteristic, the real-complete space X is a closed subspace of the product n{ ( Ry,/ 6 C(X) } of the set of copies Rf of the real line R ([2]).
"I—r f
We denote by U the uniformity on X induced by the product n{ (¿r ,/ 6 C (X) } the set of natural uniformities of the real line . The uniform space (X, U) is complete as a closed subspace of the complete of the uniform space n{( Ry,£j?) :/ 6 C (X) } . The uniformity U is generated by the restriction family by the projection p ryn{R y:/ 6 C(X) } — R A Since p ry 6 C (X) , for each / 6 C (X) , then // £ //F. Hense, i/F is the complete functional uniformity on X. Then the topology of the space X is also determined by this maximal functional uniformity .
Definition 3. A uniform space (X, U) is called a pre-maximal functionally uniform space if its completion (X, U) is uniformly real complete and uniformity U is an maximal functional uniformity.
Let X be an arbitrary Tychonoff space. Now we construct real complete extensions of the Tychonoff space by means of uniform structures.
We denote by V (X) the set of all pre-maximal uniformities of the Tychonoff space X. The sets V (X) are partially ordered by inclusion. We denote by H (X) the set (identifying the equivalent extension) of all real complete extensions of the Tychonoff space X. The set H (X) is also partially ordered in a natural way [2], [3].
On every real complete extension HX of the Tychonoff space X, there exists a unique complete maximal functional uniformity It induces on X the pre-maximal functional uniformity 0 E V (X). Each uniformity corresponds to a unique real complete extension ( Hu,X) obtained as a completion of the uniform space (X, U). It is easy to see that this correspondence between the partially ordered sets H (X) and V (X) preserves a partial order.
So, we have obtained the following theorem
Theorem 2. Partially ordered sets V (X) and H (X) are isomorphic.
Lemma. Every uniformly real complete space is uniformly homeomorphic to a closed subspace of the product of some set of copies of a real line with natural uniformity.
Theorem 3. For each uniform space (X, U) there is exactly one (up to a uniform homeomorphism) uniformly real-complete space ( with the following properties:
(1) There is a uniformly homeomorphic enclosure i:(X,UF) — (dyX.dy), for which is the completion of the uniform space , where is the maximum functional
uniformity contained in U.
(2) For any continuous function f :(X,U) — (R,ER), there is a uniformly continuous function f: (t9uX, fiu) — (R, ER) such that f i = f .
The spaces also satisfy the condition:
(3) For each uniformly continuous mapping f: (X, U) ^ (y, M) of the uniform space (X, U) into an arbitrary uniformly real complete space (y, M), there is a uniform mapping X (fi uX, fi tj) — ( y , M) such that / i=f.
A uniformly real complete space (du,X,du~) is called the Hewitt completion of the uniform space (X, U). Generally, it differs from the completion (X, U) of the uniform space (X,U).
Consider Lindeloff extensions on Tichonoff and uniform space.
Definition 4. Let (X, U) be a uniform space. The uniformity U is called:
1) preparacompact if every cover y of the set X such that y^F ^ 0 for any Fe^(U) belongs to U ;
2) strongly preparacompact if U is a preparacompact and has a base consisting of a star-finite coverings;
3) preLindeloff if U is a preparacompact and has a base consisting of countable coverings.
We denote by UD (X) (respectively UP (X), US (X), UL (X)) the set of all preuniversal (respectively preparacompact, strongly preparacompact, preLindeloff) uniformities of the Tychonoff space X . The sets UD (X), UP (X), US (X) are partially ordered by inclusion.
Theorem 4. For any Tychonoff space X the following partially ordered sets
1) (D (X),<) and (Ud (X), c);
2) (P (X),<) and (Up (X), c);
3)( S (X) ,<) and (Us (X), c);
4) (L (X) ,<) and (Ul (X), c).
are isomorphic.
Proposition 3. A Tychonoff space X is locally compact and paracompact (respectively Lindeloff) if and only if it contains a universal uniformity U* contains a cover (countable cover respectively) consisting of compact subsets.
Remark 1. Let (X, U) -be a uniform space, and (X, U) - be its completion. If (A,UA ) is a
precompact subspace of the space (X, U), then the subspace ([A]^, U^ ) of the space (X, U) is compact.
Theorem 5. There is an isomorphism between the partially ordered the set of all locally compact paracompact (locally compact Lindeloff) extensions of the given Tychonoff space X and partially ordered set of all preuniversal uniformities of the space X containing a uniform cover (respectively countable uniform cover) consisting of precompact subsets.
A partially ordered set (D (X), <) has the greatest element. This element is the extension t * ^ X corresponding to the universal uniformity U* of the space X . The rest partially ordered sets (P (X), <), (P (X), <) and (L (X), <) generally speaking, do not have greatest elements. Theorem 6. For Tychonoff space X the following conditions are equivalent:
(1) Partially ordered set (P (X), <) has a greatest element.
(2) Universal uniformity U* of the space X is preparacompact.
If is a universal (the maximal) uniform of a Tychonoff space , then a maximal locally compact paracompact (locally compact Lindeloff, respectively) extension of a uniform space is a maximal locally compact paracompact (maximal locally compact Lindeloff,
respectively) extension of the Tychonoff space .
From the above results, one can get the following theorem.
Theorem 7. Among all locally compact paracompact (locally compact Lindeloff, respectively) extensions of a Tychonoff space X there is a maximal extension.
Let /¿X be a maximal Dieudonne complete extension of the space X and p X $ ( s p X, /X) a maximal locally compact paracompact (a maximal locally compact strongly paracompact, a maximal locally compact Lindeloff respectively) extension of the space X . Then we get the following inclusions
If vX$ is a maximal real compact Hewitt extension of a space X, then the following inclusions hold.
Remark 2. Locally compact paracompact space is strongly paracompact. The locally compact strongly paracompact extensions coincide with locally compact paracompact extensions
Bibliography
1. Edwin Hewitt. Rings of real-valued continuous functions I. Transactions of the American Mathematical Society 64(1948). - P. 45-99.
2. Borubaev A.A. Ravnomernaya topologiya. Bishkek. Izdatelstvo "Ilim" 2013.
3. Aparina L.V. Hewitovskie rasshireniya blizostnych I ravnomernych prostranstv // Sibirskii mat. journal. -1974. - T.15. - №4. - P. 707-729.
4. Engeiking R. General topology. Moskow.Mir, 1986. P.752
5. Borubaev A.A. On completions of uniform spaces and extensions of topological spaces. Bolg.Math.Spis., 1989, V.15.1. P. 63-73. (in Russian)
