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Let $X$ be infinite, $\tau_X$ be the co-finite topology.

Questions:

(a) What are the connected subsets of $X$?

(b) Are there any infinite subsets of $X^2$ which are disconnected?

Here are my attempts, and I am not confident that they are correct.

Feel free to give comments and share your thoughts please.

Attempts:

(a) The closed sets $\mathcal{C}_X$ of $X$ are precisely the finite sets or $X$. Since the closed sets of $A \subseteq X$ are precisely $A \cap U$ for some $U \in \mathcal{C}_X$, we see that $A$ is connected if and only if $A$ is infinite.

(b) The collection of all intersections of sets in $\mathcal{C}_X \times \mathcal{C}_X$ gives the set of closed sets $\mathcal{C}_{X \times X}$ for $X \times X$. Thus, one can't have an infinite subset $U \times V \subseteq X \times X$ that is a disjoint union of two closed sets in $U \times V$.

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  • $\begingroup$ For (a), the subspace topology on a finite $A\subset X$ is discrete, so a finite $A$ is disconnected unless $A$ is empty or has just one member. The empty space and any $1$-point space are connected spaces. $\endgroup$ – DanielWainfleet Apr 4 '18 at 3:59
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Every infinite set with the cofinite topology is
hyperconnected, ie the intersection of two not empty
open sets is not empty. It is definitely connected.

Every infinite subset of a cofinite space is a cofinite
space, thus hyperconnected. Finite sets of a cofinite space
are discrete.

Consequently, the connected subsets of a cofinite space
are all infinite subsets, all singletons and the empty set.

Your proof of this is unconvincing.

Let a,b be two elements of an infinite cofinite space S.
Show {a,b}×S is an infinite disconnected subset of S×S.

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  • $\begingroup$ +1 especially for your example of an infinite disconnected subset of $S^2$ $\endgroup$ – DanielWainfleet Apr 4 '18 at 15:16

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