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Firstly, I will give the definition of perfect mapping:

Let $f$ be a closed mapping from a topological space $X$ to another topological space $Y$. We call it a perfect mapping if for every point $y \in Y$, the subspace $f^{-1}(y)$ of $X$ is compact.

My question is this: If $X$ has countable base, i.e., $\mathcal W(X) = \omega$, and $f$ is a perfect mapping from $X$ to $Y$, how to show that $Y$ also has countable base?

Thanks ahead:)

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1 Answer 1

up vote 5 down vote accepted

I'll copy here a proof of a more general result from the book Engelking, Ryszard (1989): General Topology. Heldermann Verlag, Berlin. ISBN 3885380064.


Theorem 3.7.19. If there exists a perfect mapping $f \colon X \to Y$ onto a space $Y$, then $w(Y) < w(X)$.

Here $w(X)$ is the weight of a topological space, i.e. $$w(X)= \min \{|\mathcal B|; \text{$\mathcal B$ is a base for $X$}\}+\aleph_0.$$

Proof. Let $w(X) = \mathfrak m$. Since the validity of our theorem is obvious for $\mathfrak m < \aleph_0$, we can assume that $\mathfrak m \ge \aleph_0$. Let $\{U_s\}_{s\in S}$ be a base for $X$ such that $|S| = \mathfrak m$ and let $\mathcal T$ be the family of all finite subsets of $S$. Since $|\mathcal T| = \mathfrak m$ it suffices to show that the family $\{W_T\}_{T\in\mathcal{T}}$, where $W_T = Y \setminus f(X \setminus \bigcup_{s\in T} U_s)$, is a base for $Y$. It follows from the definition that the sets $W_T$ are open. Let us take a point $y \in Y$ and a neighbourhood $W \subset Y$ of $y$. The inverse image $f^{-1}(y)$ is a compact subset of $f^{-1}(W)$; thus there exists a $T \in \mathcal T$ such that $f^{-1}(y)\subset \bigcup_{s\in T} U_s \subset f^{-1}(W)$. Clearly $y \in W_T$, and since $$Y\setminus W = f(X\setminus f^{-1}(W)) \subset f(X\setminus \bigcup_{s\in T} U_s)$$ we have $W_T\subseteq W$.


The above is taken verbatim from Engelking's book. But in the place where he writes $w(X)<\aleph_0$, he probably means $|S|<\aleph_0$. (Since $w(X)$ is infinite by definition.)

If a topological space has a finite base, then the topology is finite. The map $f$ is quotient, since every surjective closed map is quotient. This means that for every open set $U\subseteq Y$ the preimage $f^{-1} (U)$ is open. The assignment $U\mapsto f^{-1}(U)$ is an injection from the topology of $Y$ to the topology of $X$. Hence $Y$ has, in this case, only finitely many open sets. (And thus it has finite base.)

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Thanks Martin Slezk for your answer with a nice result. I'm surperising with somebody could think out $\{W_T\}$ is the open base of space $Y$. And, If the weight of the space $X$ is finite, does it show $X$ is a discrete space? –  Paul Jul 23 '12 at 11:03
    
Every indiscrete space has finite base. However, we usually consider $w(X)=\aleph_0$ even in the case that there is a finite base. –  Martin Sleziak Jul 23 '12 at 11:06
    
I see. Could you say more about the thing when the $m < \aleph_0$? I unstand this a little. Thanks again. –  Paul Jul 23 '12 at 11:12
    
@Paul I've added some clarification for that case that $X$ has finite base. –  Martin Sleziak Jul 23 '12 at 12:22
    
Thank you. I've seen the added part. I'll accept your answer. –  Paul Jul 24 '12 at 10:17

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