TITLE: compact subset of the product space $X\times X$
QUESTION [0 upvotes]: Suppose $X$ is a topological space,consider the product space $X\times X$.can we deduce that any compact subset of $X\times X$ is the form of $K\times K$,where $K$ is the compact subset of $X$.

REPLY [1 votes]: No. Consider $[0,1]\times [0,2]$ in  $[0,2]\times[0,2]$.
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