[Math] composition of functions is continuous

compactnesscontinuitygeneral-topology

Question is as follows : Let $X,Y,Z$ are metric Spaces

Let $f:X\rightarrow Y$ be continuous map onto $Y$ and let $X$ be compact. Also $g:Y\rightarrow Z$ such that $g\circ f:X\rightarrow Z$ is continuous. Show that $g$ is continuous.

I do not even know how to start this question..

Please give some hints.. [Not looking for nor wanting a full answer]

Do i have to check that inverse image of open set is open under $g$ or is there any other methods?

$U$ is open in $Z$ so, $(g\circ f)^{-1}(U)$ is open.. i.e., $f^{-1}(g^{-1}(U))$ is open… I do not think this is a better way to proceed…

Best Answer

Thanks for DanielFischer's comment. The following assumes $Y$ is Hausdorff in order to make point 5 to be true.

Hints:

  1. Start with closed set $C$ in $Z$. There is an equivalent definition for continuity of $g$ with respect to closed sets.

  2. By surjective of $f$, we know $f(f^{-1}(A))=A$ for any set $A$ in $Y$.

  3. Use continuity of $g\circ f$ and $f$.

  4. Note closed subsets in a compact space is compact.

  5. Note image of any compact set under continuous function is compact , which is also closed .

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