Excluded point topology

Example

Let $X$ be a set and let $p \in X$. The excluded point topology on $X$ declares a set $U \subseteq X$ open if $U = X$ or $p \notin U$. This is a topology: $\emptyset$ and $X$ are open. A union of open sets is either $X$ or avoids $p$, and so is a finite intersection of open sets. So $X$ is a topological space.

• The closed sets are $\emptyset$ and the sets that contain $p$.
• $X$ is compact: the only open set that contains $p$ is $X$, so every open cover has $X$ as a member. Then $\set{X}$ is a finite subcover.
• $X$ is connected: a clopen set that contains $p$ is open, hence equal to $X$. A clopen set that misses $p$ is closed, hence empty.
• $X$ is a T0 space: for $x \ne p$, the open set $\set{x}$ contains $x$ but not $p$. For distinct $x, y \ne p$, the open set $\set{x}$ separates them.
• If $X$ has at least two points, $X$ is not a T1 space, since every open set that contains $p$ contains all points.

The closed sets of this topology are exactly the open sets of the particular point topology at $p$.