Fort Space is Excluded Point Space with Finite Complement Space

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$.


Then $\tau_p$ is the minimal topology that is generated by the excluded point topology and the finite complement topology.


Proof

Let $T_1 = \struct {S, \tau_1}$ be the excluded point space on $S$ from $p$.

Let $T_2 = \struct {S, \tau_2}$ be the finite complement space on $S$.


By definition:

$\tau_1 = \set {H \subseteq S: p \in \relcomp S H} \cup \set S$
$\tau_2 = \leftset {H \subseteq S: \relcomp S H}$ is finite$\rightset {} \cup \set \O$


By definition of Fort space, we have:

$U \in \tau_1 \implies U \in \tau_p$
$U \in \tau_2 \implies U \in \tau_p$

So $\tau_1 \cup \tau_2 \subseteq \tau_p$.

Similarly:

$U \in \tau_p \implies U \in \tau_1 \lor U \in \tau_2$

and so $\tau_p \subseteq \tau_1 \cup \tau_2$.

So $\tau_p = \tau_1 \cup \tau_2$ and the result follows from Union is Smallest Superset.

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Fort_Space_is_Excluded_Point_Space_with_Finite_Complement_Space&oldid=470683"