Open Sets of Double Pointed Topology/Corollary
Theorem
Let $\struct {S, \tau_S}$ be a topological space.
Let $D$ be a doubleton endowed with the indiscrete topology.
Let $\struct {S \times D, \tau}$ 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 $\map \complement X$ is open.
By Open Sets of Double Pointed Topology, it follows by that for some $U \in \tau$:
- $\map \complement X = U \times D$
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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- $X = \paren {S \times D} \setminus \paren {U \times D} = \paren {S \setminus U} \times D$
Since $S \setminus U$ is a closed set of $\tau$, the result follows.
$\blacksquare$