Closure of Subset of Double Pointed Topological Space

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 $S$.

Let $X \subseteq S \times D$ be a subset of $S \times D$.

Then the closure of $X$ in $\tau$ is:

$\map \cl X = \map \cl {\map {\pr_1} X} \times D$

where $\pr_1$ denotes the first projection on $S \times D$.

Proof

By Closed Sets of Double Pointed Topology, $\map \cl {\map {\pr_1} X} \times D$ is closed in $\tau$.

Furthermore, for $\tuple {s, d} \in X$, one has:

$s \in \map {\pr_1} X \subseteq \map \cl {\map {\pr_1} X}$

by definition of closure.

Since also $d \in D$, we conclude that:

$X \subseteq \map \cl {\map {\pr_1} X} \times D$

By $(3)$ of Equivalence of Definitions of Topological Closure, it now suffices to prove that:

This article, or a section of it, needs explaining.
In particular: Check the link above -- probably better to link directly to the specific definition of Closure, and also below
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.