Some characterization of finite complement topology Let $Y:=\{(u,v)\in \mathbb{R}^2: u^2+v^2\le r\}$ where $r>0$ is finite complement topology.
Now, If I want to characterize all the connected subsets of $Y$, then how should I start for such characterization?
I don not know how to formally write it.
Any help is highly appreciated.
FYI, I am learning it for the first time.
 A: First of all the finite complement topology depends only on the cardinality of the underlying set. Meaning that it doesn't matter if you consider it on $\mathbb{R}$ or your $Y$ or $[0,1]$. The finite complement topology makes them all homeomorphic.
Now assume that $X$ is an infinite set with the finite complement topology. Let $U$, $V$ be two nonempty and open subsets. By the definition $X\backslash U$ and $X\backslash V$ are finite. But since $X$ is infinite then there is $x_0\in X$ such that $x_0\not\in X\backslash U$ and $x_0\not\in X\backslash V$. Or equivalently $x_0\in U$ and $x_0\in V$. And thus any two nonempty open subsets in $X$ have nonempty intersection. In particular $X$ is connected.
On the other hand a finite set $X$ with finite complement topology is discrete. And the only discrete connected space is the singleton $\{*\}$.
Finally, every subspace of a finite complement topological space has finite complement topology. Which leads to:

A subset of a finite complement topological space $X$ is connected if and only if it is a singleton $\{*\}$ or infinite.

