I know that separate continuity does not, in general, imply joint continuity but does the converse hold? Given $X,Y,Z$ topological spaces, if $f: X\times Y \to Z$ is continuous ($X\times Y$ with the product topology) does it follow that $f$ is separately continuous?

  • $\begingroup$ Is this your question : "Is $f_{X,y_0}:{X\times \{ y_0\} }\to Z$ continuous ? " $\endgroup$
    – Jo Jomax
    Apr 9 '20 at 15:13
  • $\begingroup$ @JoJomax hey, i think you mistyped it because it is not in mathematical format but i think it is! $\endgroup$
    – MathMath
    Apr 9 '20 at 15:15
  • $\begingroup$ Yes . $f_{|_{X\times \{ y_0\} }}$ is continuous . $\endgroup$
    – Jo Jomax
    Apr 9 '20 at 15:17

Yes, if $f : X \times Y \to Z$ is continuous, then for each $(x_0, y_0) \in X \times Y$, $f(x_0, y)= f\restriction_{\{x_0\} \times Y}$ is continuous as a map from $Y \to Z$ and so is $f(x, y_0) = f\restriction_{ X \times \{y_0\}}$ as a map $X \to Z$. The maps $x \to (x,y_0)$ and $y \to (x_0, y)$ are always embeddings.


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