If we have topological space $\mathbb{R}$ equipped with the co finite topology. If we have finite subsets in consideration they are definitely closed because open sets are of the form complement of finite set.
If we have infinite or finite subsets of real line in consideration how to look at their compactness and connectedness?
Best Answer
Let $X$ be a set and equip it with the cofinite topology. Let $Y$ be a non-empty subset of $X$. We want to show that the relative topology on $Y$ is again the cofinite topology.
An open set in $Y$ has the form $V=U\cap Y$, where $U$ is open in $X$; then $$ Y\setminus V=Y\cap V^c=Y\cap(U\cap Y)^c= Y\cap(U^c\cup Y^c)=Y\cap U^c $$ (where $A^c$ means the complement of $A$ in $X$). Since $U^c$ is closed in the cofinite topology (in $X$), either $U^c=X$ or $U^c$ is finite. Therefore $Y\setminus V$ is either $Y$ or it is finite.
Since $Y\ne\emptyset$ has the cofinite topology, it is compact. Take an open cover, choose a non-empty element and what's missing from it is closed, so it is covered by a finite number of open sets in the cover.
If it is infinite it is also connected: if $Y=A\cup B$, with $A$ and $B$ open and disjoint, one of them must be empty, because they are also closed.