Past exam of the options II course of the University of Oxford / 2005 / E / 3 / Solution

The product topology on $X\times Y$ has the open rectangles $U\times V$, with $U$ open in $X$ and $V$ open in $Y$, as a topological basis. The projection preimages are $p_X^{-1}(U)=U\times Y$ and $p_Y^{-1}(V)=X\times V$, so both projections are continuous.

If $f:Z\to X\times Y$ is continuous, both coordinate compositions are continuous. Conversely, if both are continuous, thenis open for every basic rectangle, proving continuity of $f$. Given paths between two chosen endpoints in each factor, their coordinatewise pair is therefore a continuous path in the product, proving that a product of path-connected spaces is path-connected.

For the compact-neighborhood assertion, fix $x\in K$. For each $y\in L$, choose an open rectangle $U_y\times V_y$ containing $(x,y)$ inside $W$. Compactness of $L$ chooses finitely many $V_{y_j}$ covering $L$. PutThen $x\in U_x$, $L\subseteq V_x$, and $U_x\times V_x\subseteq W$: any point of $V_x$ belongs to one of those rectangles' second factors, while $U_x$ lies in every first factor.

Compactness of $K$ now selects finitely many $U_{x_i}$ covering $K$. SetThese are open, contain $K$ and $L$, and if $(u,v)\in U\times V$, choose an $i$ with $u\in U_{x_i}$ and use $v\in V_{x_i}$. HenceIf $K$ or $L$ is empty, take the corresponding open factor empty and the other factor the whole space; the assertions then hold directly. No Hausdorff assumption on the factors is needed.