Let $f:X\times Y\mapsto R$ be jointly continuous, where $X$ is some topological space, $Y$ is some countably compact topological space. My question is whether or not the envelop $\phi(x) := \max\_{y\in Y} f(x,y)$ is continuous?


I know the answer is yes, if we strengthen countably compact to compact. And I tried to construct a counterexample with $Y = [0, \aleph\_1)$ but failed. Any idea will be appreciated!



Continuity of the maxima
In general the answer is no. For showing this claim I must bring two theorems. You could find these theorems in the page 238 of the text "**General topology**" written by **Ryszard Engelking**.


The following theorem is due to **Isiwata, Nobel, Hager and comfort**:


**Theorem1**: For the Tychonoff spaces $X , Y$ the following are equivalent:


* The Projection $p:X\times Y \rightarrow X$ maps zero-sets of $X\times Y$ to closed sets of $X$.
* Every bounded continuous function $f:X\times Y \rightarrow \mathbb{R}$ can be continuously extended over $X \times \beta Y$.
* For every bounded continuous function $f:X\times Y \rightarrow \mathbb{R}$ the formula $F(x)=sup\_{y\in Y}f(x,y)$ defines a continuous function $F:X\rightarrow \mathbb{R}$.


The following theorem is due to "**Tamano**" which is essential for our claim:


**Theorem2**: The cartesian product $X\times Y$ of Tychonoff spaces $X , Y$ is pseudocompact if and only if $X$ and $Y$ are pseudocompact and The Projection $p:X\times Y \rightarrow X$ maps zero-sets of $X\times Y$ to closed sets of $X$. 


But for showing our claim as you probably Know there is a countably compact space $X$ which the product space $X\times X$ is not even Pseudocompact.(You could find such example in the chapter9 of the text **Rings of continuous functions** written by **Gillman and jerison**)


Then because $X$ is countably compact it is also pseudocompact. and because $X\times X$ is not pseudocompact (**as we mentioned above**) each of the statements in Theorem1 fails. Then for example there is a bounded continuous function $f:X\times X \rightarrow \mathbb{R}$ so that the function $F(x)=sup\_{y\in Y}f(x,y)$ is not continuous.


