Open Sets of Double Pointed Topology/Corollary
Theorem
Let $\left({S, \tau}\right)$ be a topological space.
Let $D$ be a doubleton endowed with the indiscrete topology.
Let $\left({S \times D, \tau}\right)$ be the double pointed topology on $S$.
A subset $X \subseteq S \times D$ is closed in $\tau$ if and only if for some closed set $C$ of $\tau$:
- $X = C \times D$
Proof
By definition, $X$ is closed if and only if its complement $\complement \left({X}\right)$ is open.
By Open Sets of Double Pointed Topology, it follows by that for some $U \in \tau$:
- $\complement \left({X}\right) = U \times D$
Then by Cartesian Product Distributes over Set Difference and Complement of Complement, we have that:
rather, Cartesian Product Distributes over Complement (and for Relative Complement), but those don't exist
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- $X = \left({S \times D}\right) \setminus \left({U \times D}\right) = \left({S \setminus U}\right) \times D$
Since $S \setminus U$ is a closed set of $\tau$, the result follows.
$\blacksquare$
