Basic Topological problem on closed sets

Question

How can one prove this proposition:

proposition 5.15: $f$ is continuous iff $f(\overline{A}) \subset \overline{f(A)}$, where $\overline{A}$ denotes the closure of a set $A.$

It seems to be a small result, but on the chapter on continuity, it just states this proposition, but I have little clue on how to show this is true. I would appreciate the help.

@rbird Thank you very much for the source, I will try to study the proof. — Aurora Borealis, Mar 02 '18 at 14:41

score 1 · Accepted Answer · answered Mar 02 '18 at 14:42

1

Let, $f:(X,\tau)\rightarrow(Y,\tau')$ be continuous, then for any closed set $F$ in $(Y,\tau')$ $f^{-1}(F)$ is closed in $X$.

Now let $A\subset X$ be any set. Then $\overline{f(A)}$ is a closed set in$(Y,\tau')$ also $f^{-1}(\overline{f(A)})$ is closed in $(X,\tau)$ with, $A\subset f^{-1}(\overline{f(A)})$.

But, $\overline{A}$ is the smallest subset containing $A$. So, $\overline{A}\subset f^{-1}(\overline{f(A)})$ i.e. $f(\overline{A})\subset \overline{f(A)}$.

answered Mar 02 '18 at 14:42

Sujit Bhattacharyya

3,030

That is only one side of it. – drhab Mar 02 '18 at 14:44
@drhab - did I make any mistake? please edit if necessary – Sujit Bhattacharyya Mar 02 '18 at 14:46
You did not make a mistake, but what lacks in your answer is a proof of the other side: if $f(\overline A)\subseteq\overline{f(A)}$ for every $A\subseteq X$ then $f$ is continuous. – drhab Mar 02 '18 at 14:49
oh! yes. sorry for that – Sujit Bhattacharyya Mar 02 '18 at 16:17

Basic Topological problem on closed sets

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