Question

5. Let R be equipped with the Fort topology TFp, defined by TF,-(U C R l p ¢ U or R-U is finite), with pER a point. (i) Is (R, (i) Is (R, TFp) path-connected? TF) connected?

Answer 1

i) Let $q\in\mathbb R$ be such that $q\neq p$. Then $U:=\mathbb R-\{q\}$ is open in Fort topology since $\mathbb R-U=\{q\}$ is finite. Also, $V:=\{q\}$ is open in Fort topology since $p\notin V$. Thus, we have found non-empty disjoint open subsets $U,V\subseteq \mathbb R$ such that $U\cup V=\mathbb R$ . Hence, is not connected.

R, TF.

ii) Since path-connected spaces are necessarily connected and is not connected, it can not be path-connected either.

R, TF.