Rudin's PMA Theorem 4.8

Asked Apr 20 '22 at 08:15

Active Apr 20 '22 at 08:15

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Here is rudin's statement

Theorem 4.8 A mapping $f$ of a metric space $X$ into a metric space $Y$ is continuous if and only if $f^{-1}(V)$ is open in $X$ for every open set $V$ in $Y$.

Theorem 4.8 (corollary) A mapping $f$ of a metric space $X$ into a metric space $Y$ is continuous if and only if $f^{-1}(C)$ is closed in $X$ for every closed set $C$ in $Y$.

This follows from the theorem, since a set is closed if and only if its complement is open, and since $f^{-1}(E^{c})=[f^{-1}(E)]^{c}$ for every $E \subset Y$

Here is my question

We know it is $X$ mapping into $Y$, then $f(X)\subset Y$, maybe Y contains some elements $\notin f(X)$

I want to know should these elements be considered in $V$ and $C$?

asked Apr 20 '22 at 08:15

Sure, $Y$ may contain some elements $\notin f(X)$ but what is the problem with that? – Kavi Rama Murthy Apr 20 '22 at 08:18
@Kavi Rama Murthy For example, C is a closed set which contains some of such elements, does $f^{-1}(C)$ have some meanings when applies for these elements? – Apr 20 '22 at 08:24
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If $f: X \to Y$ and $A \subseteq Y$ then $f^{-1}(A)$ is defined as ${x \in X : f(x) \in A}$. This always makes sense. (It may be the empty set). – Kavi Rama Murthy Apr 20 '22 at 08:27
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The inverse image of $C$ is a union of all preimages of each element, Inverse images of such elements will be the empty set. – ganesh Apr 20 '22 at 08:29
@ganesh Thanks! – Apr 20 '22 at 08:31
@Kavi Rama Murthy Thanks! – Apr 20 '22 at 08:33

Rudin's PMA Theorem 4.8

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