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UDC 513.831
MATHEMATICS
V. FEDORCHUK
θ-SPACES AND PERFECT IRREDUCIBLE MAPPINGS OF TOPOLOGICAL SPACES
(Presented by Academician P. S. Aleksandrov on 25 VII 1966)
In the present paper the notion of a θ-space is introduced, generalizing the notion of a proximity space compatible with a completely regular space.
At the same time, θ-spaces exist on every regular space, whereas proximity spaces are compatible only with completely regular spaces. A connection is given between θ-spaces on a regular space \(X\) and bicompact extensions of all its perfect irreducible preimages. In this connection, a theorem of Yu. M. Smirnov \((^4)\) on the one-to-one correspondence between proximity spaces compatible with a given completely regular space \(X\) and bicompact extensions of the space \(X\) is generalized. The paper also introduces the notion of a θ-mapping of θ-spaces and studies its connection with mappings of bicompact extensions of perfect irreducible preimages of topological spaces.
Let \(X\) be a regular space. We shall say that a θ-proximity is given on \(X\) if for any two subsets \(A \subset X\) and \(B \subset X\), either \(A \theta B\) or \(A \bar{\theta} B\) is specified and the following axioms are satisfied:
I. \(A \theta B \Rightarrow B \theta A\).
II. \(A \bar{\theta} B_i,\ i = 1, 2 \Leftrightarrow A \bar{\theta}(B_1 \cup B_2)\).
III. \(\varnothing \bar{\theta} X\).
IV. \(\{x\}\theta A \Rightarrow x \in [A]\).
V. \(A \bar{\theta} B \Rightarrow\) there exists such a \(C=\langle [C]\rangle \supset A\) that \(A \bar{\theta}(X \setminus [C])\) and \(C \bar{\theta} B\) *.
Axioms I–IV coincide with the corresponding axioms of a proximity space compatible with the given topological space, and it is easy to see that axiom V is a weakening of the normality axiom of a proximity space compatible with a topological space.
An example of a θ-proximity on a regular space \(X\) is the following relation: \(A \bar{\theta}_a B \Leftrightarrow\) there exist disjoint neighborhoods of the sets \(A\) and \(B\). It is easy to verify that all the axioms are satisfied. It will be shown below that the θ-proximity \(\theta_a\) is maximal.
Theorem 1. Let \(f: Z \to X\) be a perfect irreducible mapping of a completely regular space \(Z\) onto a space \(X\), and let \(bZ\) be a bicompact extension of the space \(Z\). Then the bicompactum \(bZ\) generates on \(X\) the following θ-proximity:
\[ A \bar{\theta} B \Longleftrightarrow [f^{-1}A]_{bZ} \cap [f^{-1}B]_{bZ} = \varnothing . \]
We shall call a regular space \(X\) with a θ-proximity given on it a θ-space.
* By \(\langle D\rangle\) is denoted the interior of the set \(D\).
The main result of the paper is
Theorem 2. Every $\theta$-space on a regular space $X$ determines a completely regular space $X_\theta$, a perfect irreducible projection $\pi_{X\theta}: X_\theta \to X$ of the space $X_\theta$ onto $X$, and a bicompact extension $b_\theta X_\theta$, which generates the given $\theta$-proximity.
Remark. The space $b_\theta X_\theta$ is the space of maximal centered systems $\tau=\{H\}$ of open subsets $H$ of the space $X$ with the following $\theta$-regularity condition: for every $H\in\tau$ there exists an $H'\in\tau$ such that $H'\,\theta\,(X\setminus[H])$*.
Theorem 3. Let $f_1: Z_1\to X$ and $f_2: Z_2\to X$ be perfect irreducible mappings of completely regular spaces $Z_1$ and $Z_2$ onto $X$. If the bicompact extensions $b_1Z_1$ and $b_2Z_2$ generate on $X$ the same $\theta$-space, then there exists a homeomorphism $g: b_1Z_1\to b_2Z_2$ such that $gZ_1=Z_2$ and $f_1=f_2g$.
Corollary 1. There exists a one-to-one correspondence between the partially ordered set of all $\theta$-spaces on a regular space $X$ and the set of all pairs $(bZ, f: Z\to X)$, where $bZ$ is a bicompact extension of the space $Z$ and $f$ is a perfect irreducible mapping onto $X$*.
Corollary 2. There exists a maximal $\theta$-space on a regular space $X$.
Corollary 3. For any two perfect irreducible mappings $f_1: Z_1\to X$ and $f_2: Z_2\to X$ onto $X$ there exists a perfect irreducible mapping $f: Z\to X$ onto $X$ and perfect irreducible mappings $g_1: Z\to Z_1$ onto $Z_1$ and $g_2: Z\to Z_2$ onto $Z_2$ such that $f_1g_1=f=f_2g_2$, and moreover the mapping $f$ is minimal in the sense that mappings $f'$, $g_1'$ and $g_2'$ with the same relations factor through the mappings $f$, $g_1$ and $g_2$, respectively**.
S. Iliadis proved [2] that the space $B$ of maximal centered systems of open sets of a regular space $X$ is the Čech extension of the absolute of the space $X$. A maximal centered system of open sets in a regular space $X$ obviously satisfies the $\theta$-regularity condition for the $\theta$-proximity $\theta_a$. Hence, by the remark to Theorem 2 and Corollary 1, $\theta_a$ is the maximal $\theta$-proximity.
Definition. A perfect irreducible mapping $f: X\to Y$ of a $\theta$-space $X$ onto a $\theta$-space $Y$ will be called a $\theta$-mapping if $A,B\subset Y$ and
\[ \overline{A}\,\theta\,B\Rightarrow \langle f^{-1}[A]\rangle\,\overline{\theta}\,\langle f^{-1}[B]\rangle . \]
Theorem 4. Every $\theta$-mapping $f: X\to Y$ generates a perfect irreducible mapping $f_\theta: b_\theta X_\theta\to b_\theta Y_\theta$ such that $f_\theta X_\theta=Y_\theta$ and $f\pi_{X\theta}=\pi_{Y\theta}f_\theta$.
In accordance with (3), we shall call a multivalued mapping $f: X\to Y$ of $\theta$-spaces a multivalued $\theta$-mapping if there exists a $\theta$-space $Z$ and $\theta$-mappings $f_X: Z\to X$, $f_Y: Z\to Y$ such that $f=f_Y f_X^{-1}$.
* The notion of a centered system of open sets with a certain regularity condition was introduced by P. S. Aleksandrov [1] in the construction of the maximal bicompact extension.
** The order in this set is analogous to the order in the set of proximity spaces compatible with the given space.
*** Two pairs $(bZ, f: Z\to X)$ and $(b'Z', f'Z'\to X)$ are identified if there exists a homeomorphism $g: bZ\to b'Z'$ such that $gZ=Z'$ and $f=f'g$. $(bZ,f)\geq (b'Z',f')\Longleftrightarrow$ there exists a perfect irreducible mapping $g: bZ\to b'Z'$ such that $gZ=Z'$ and $f=f'g$. It is easy to verify that pairs $(bZ,f)$ and $(b'Z',f')$ for which both inequalities $(bZ,f)\leq (b'Z',f')$ and $(bZ,f)\geq (b'Z',f')$ hold are identified.
**** A mapping $f': Z'\to X$ factors through a mapping $f: Z\to X$ if there exists a mapping $g: Z'\to Z$ such that $f'=fg$.
Theorem 5. Every multivalued \(\theta\)-mapping \(f:\ X \xleftarrow{f_X} Z \xrightarrow{f_Y} Y\) generates a multivalued perfect irreducible mapping
\[ f_\theta:\ b_\theta X_\theta \xleftarrow{\varphi} b_\theta Z_\theta \xrightarrow{\psi} b_\theta Y_\theta, \]
such that \(\varphi^{-1}X_\theta=\psi^{-1}Y_\theta\) and \(f\pi_{X\theta}=\pi_{Y\theta}f_\theta\).
Theorem 6. Let \(\theta\)-spaces \(X\) and \(Y\) be given. Then for every multivalued perfect irreducible mapping \(f_\theta:\ b_\theta X_\theta \xleftarrow{\varphi} B \xrightarrow{\psi} b_\theta Y_\theta\) satisfying \(\varphi^{-1}X_\theta=\psi^{-1}Y_\theta\), there exists a multivalued \(\theta\)-mapping \(f:\ X \xleftarrow{f_X} Z \xrightarrow{f_Y} Y\) such that
\[ B=b_\theta Z_\theta,\qquad f\pi_{X\theta}=\pi_{Y\theta}f_\theta, \tag{*} \]
and, among all multivalued \(\theta\)-mappings satisfying condition \((*)\), there exists a maximal one in the factorization sense.
In conclusion I express my gratitude to my adviser P. S. Aleksandrov for his attention to my work.
Moscow State University
named after M. V. Lomonosov
Received
8 VII 1966
REFERENCES
- P. S. Aleksandrov, Matem. sborn., 5, 403 (1939).
- S. Iliadis, DAN, 149, No. 1, 22 (1963).
- V. I. Ponomarev, Matem. sborn., 51, No. 4, 515 (1960).
- Yu. M. Smirnov, Matem. sborn., 31, No. 3, 543 (1952).