I want to show that if $X$ and $Y$ are connected metric spaces, then $X \times Y$ is also connected. As a hint it is marked to write $X \times Y$ as Unions of sets of the form $(X \times \{y\}) \cup ( \{x\} \times Y)$, but i don't have an idea how to continue. Any hints?
Product of connected metric spaces
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real-analysis
general-topology
connectedness
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0ProofWiki: [Product Space is Connected iff Factors are Connected](http://www.proofwiki.org/wiki/Product_Space_is_Connected_iff_Factors_are_Connected). – 2012-07-07
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0Do you need to use the hint? It seems to me simpler to use the following characterization: $X$ is connected if and only if every continuous map from $X$ onto $\{0,1\}$ (with the discrete topology) is constant. – 2013-11-05