Every Filter has Limit Point implies Every Ultrafilter Converges
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Theorem
Let $T = \struct {S, \tau}$ be a topological space.
Let each filter on $S$ have a limit point in $S$.
Then each ultrafilter on $S$ converges to a point in $S$.
Proof
Let $T = \struct {S, \tau}$ be such that each filter on $S$ has a limit point in $S$.
Let $\FF$ be an ultrafilter on $S$.
By hypothesis, $\FF$ has a limit point $x \in S$.
By Limit Point iff Superfilter Converges, there exists a filter $\FF'$ on $S$ which converges to $x$ satisfying $\FF \subseteq \FF'$.
By Definition of Ultrafilter on Set:
- $\FF$ is an ultrafilter if and only if whenever $\GG$ is a filter on $S$ and $\FF \subseteq \GG$ holds, then $\FF = \GG$
Therefore:
- $\FF = \FF'$.
Thus $\FF$ converges to $x$.