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I was asked the following:

Are there any infinite subsets of $X \times X$ which are not connected? $X$ is given the cofinite topology, and $X \times X$ is given the product topology.

edit: $X$ is connected, as there are no disjoint nonempty open connected sets. But this holds too for the basis elements of $X \times X$ isn't it? So all such subsets are connected?

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  • $\begingroup$ Why do you think $X$ isn't connected? $\endgroup$ – Demophilus Jun 20 '17 at 23:28
  • $\begingroup$ If $X$ is infinite, $X$ is connected since a nonempty open subset is cofinite, and for it to be also closed and not the whole space, it would have to be finite which gives a contradiction. $\endgroup$ – Daniel Schepler Jun 20 '17 at 23:29
  • $\begingroup$ Base elements connected does not mean that the space is connected, just locally connected. $\endgroup$ – Henno Brandsma Jun 21 '17 at 7:47
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Note that for any $x \neq y \in X$ we have that $\{x,y\} \times X \subset X \times X$ (infinite if $X$ is) is disconnected; its connected components are exactly the sets $\{x\} \times X$ and $\{y\} \times X$. You could also use that $A \times B$ is connected iff $A$ and $B$ are connected, and all finite subsets of $X$ are disconnected (discrete).

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Yes, the product of X and any multi-point finite subset of
X, is an infinite subset of XxX that is not connected.
X itself is hyperconnected
(every intersection of two not empty open sets is not empty).

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  • $\begingroup$ How is this relevant for subsets of $X \times X$? $\endgroup$ – Henno Brandsma Jun 21 '17 at 13:22
  • $\begingroup$ @HennoBrandsma It is a generalization of your answer. $\endgroup$ – William Elliot Jun 22 '17 at 19:30

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