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George F Simmons, Topology and Modern Analysis pg.79 Problem 4

Let $X$ and $Y$ be metric spaces. Show that an into mapping $f:X \rightarrow Y$ is continuous $\iff$ $f^{-1}\left(G\right)$ is closed in $X$ whenever $G$ is closed in $Y$.

I can prove the problem for open sets, and I have been trying hard for closed. However, seems like I am stuck somewhere missing something obvious. Please don't answer directly, just give a small hint if possible.

EDIT: I am using the definition that $f^{-1}\left(G\right)$ exists only when $f$ is onto and if it is not then $f^{−1}\left(G\right)$ is a loose term for $f^{-1}\left(H\right)$ where $H$ is the range of $f$ in $G$.

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  • $\begingroup$ What is the definition of "closed" you are working with? $\endgroup$ – Zev Chonoles Jul 24 '12 at 6:25
  • $\begingroup$ Also, see this post for an answer in the more general context of topological spaces. $\endgroup$ – Zev Chonoles Jul 24 '12 at 6:27
  • $\begingroup$ Definition of closed is that the a contains all its limit points where $a$ is a limit point of $X$ if for every open sphere $S_{\epsilon}\left(a\right)$ there exists an $x \in X$ such that $x \in S_{\epsilon}\left(a\right)$ $\endgroup$ – user14082 Jul 24 '12 at 6:55
  • $\begingroup$ I just don't understand this confusion about $f^{-1}(G)$ (or $f^{-1}[G]$, as I prefer to write). For $f:X\to Y$ and $G\subseteq Y$ the definition of $f^{-1}[G]$ is simply $\{x\in X:f(x)\in G\}$. I don't see any problem with that. $\endgroup$ – Stefan Geschke Jul 24 '12 at 12:04
  • $\begingroup$ I was not taught that way, not using that or equivalent definition, I was taught as I wrote in that comment. While, I did learn the new definition, once in a while the old habit unconsciously takes over. I hope, I get used to the new one fast enough though :-) $\endgroup$ – user14082 Jul 25 '12 at 7:16
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So you can show that $f$ is continuous iff preimages of open sets are open. Now you go from there. Closed sets are complements of open sets. What is $f^{-1}[Y\setminus G]$?

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  • $\begingroup$ But the mapping is an into mapping, how do I know $f^{-1}$ exists for all Y $\endgroup$ – user14082 Jul 24 '12 at 6:52
  • $\begingroup$ It's not relevant. You can always define $f^{-1} Y$ for any subset $Y$ of the codomain of $f$. $\endgroup$ – Zhen Lin Jul 24 '12 at 6:55
  • $\begingroup$ @Jayesh: The notation $f^{-1}[Y\setminus G]$ doesn't imply that $f^{-1}$ exists for all of $Y$. It's just the set of all preimages of elements of $Y\setminus G$, not all of which necessarily have preimages. Since all of $X$ is mapped into $Y$, we have that $f^{-1}[Y]$ is all of $X$, even though $f(X)$ need not be all of $Y$. $\endgroup$ – joriki Jul 24 '12 at 6:55
  • $\begingroup$ So lets say, $Z \in Y$ is the "range" of $f$ so that $f:X \rightarrow Z$ is an onto mapping. Then, if $G$ is closed in $Y$, then $G^{-1}$ is open in $Y$, but we are now considering $H = G^{-1} \bigcap Z$, how do I know that is also open? As far as I know, $H$ is open as a subset of $Z$ (which should be a subspace of $X$) in and only if it is the intersection with $Z$ of an open set which it is, but what about $H$ in $X$? $\endgroup$ – user14082 Jul 24 '12 at 7:06
  • $\begingroup$ I guess, I am not using the definition of $f^{-1}\left(G\right)$ properly. I am using the definition that $f^{-1}\left(G\right)$ exists only when $f$ is onto and if it is not then $f^{-1}\left(G\right)$ is a loose term for $f^{-1}\left(H\right)$ where $H$ is the range. If someone one can make it more clear to me, it would be better. $\endgroup$ – user14082 Jul 24 '12 at 7:13
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The following steps lead to a solution:

(1) $G$ is closed in $Y$ iff $Y - G$ is open in $Y$.

(2) For any $A \subseteq Y$, we have $$f^{-1}(Y - A) = X - (f^{-1}(A)).$$

(3) Conclude that $f$ is continuous on $X$ $\iff$ $f^{-1}(G)$ is closed in $X$ whenever $G$ is closed in $Y$.

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  • $\begingroup$ Your second step would be true only for onto mappings, while the textbook says $f$ is an into mapping, or am I missing something? $\endgroup$ – user14082 Jul 24 '12 at 6:54
  • $\begingroup$ @Jayesh: See the comments under Stefan's answer. $\endgroup$ – joriki Jul 24 '12 at 6:56
  • $\begingroup$ @JayeshBadwaik You can define the preimage regardless of what kind of map $f$ is (injective, surjective or neither). (2) that I stated above always holds true. $\endgroup$ – user38268 Jul 24 '12 at 7:16

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