Prove that (X,d) is connected if and only if X x X isconnected. Solution Conside

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Consider the projection map X x X -> X given by (x,y) ->x. This is continuous and onto. So if X x X isconnected, then X is connected, being a continuous image of aconnected set. Conversely, suppose X is connected. Take any nonempty openand closed subset E of X x X. We want to show that E = X xX. Since it's nonempty, take some (a,b) in E. Thenthere's a map X -> X x X given by x -> (x , b). Thismap is clearly continuous, and the inverse image in X of E underthis map is again open & closed, and also nonempty (a is in theinverse image), so the inverse image of E under this map is all ofX (by connectedness). That is, if (a,b) is in E then X x {b}is a subset of E. Likewise, {a} x X is a subset of E. Now it's clear that X x X is a subset of E (just run the argumentagain), so E = X x X.

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