Metric and Topological Spaces | Part IB, 2005

Suppose that (X,dX)\left(X, d_{X}\right) and (Y,dY)\left(Y, d_{Y}\right) are metric spaces. Show that the definition

d((x1,y1),(x2,y2))=dX(x1,x2)+dY(y1,y2)d\left(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\right)=d_{X}\left(x_{1}, x_{2}\right)+d_{Y}\left(y_{1}, y_{2}\right)

defines a metric on the product X×YX \times Y, under which the projection map π:X×YY\pi: X \times Y \rightarrow Y is continuous.

If (X,dX)\left(X, d_{X}\right) is compact, show that every sequence in XX has a subsequence converging to a point of XX. Deduce that the projection map π\pi then has the property that, for any closed subset FX×YF \subset X \times Y, the image π(F)\pi(F) is closed in YY. Give an example to show that this fails if (X,dX)\left(X, d_{X}\right) is not assumed compact.

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