Closure of Open Set of Particular Point Space

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $T = \left({S, \tau_p}\right)$ be a particular point space.

Let $U \in \tau_p$ be open in $T$ such that $U \ne \varnothing$.


Then:

$U^- = S$

where $U^-$ denotes the closure of $U$.


Proof

Follows directly from:

Particular Point Topology is Closed Extension Topology of Discrete Topology
Closure of Open Set of Closed Extension Space

$\blacksquare$


Sources