# Open Sets of Double Pointed Topology/Corollary

Jump to navigation
Jump to search

## 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:

Although this article appears correct, it's inelegant. There has to be a better way of doing it.In particular: rather, Cartesian Product Distributes over Complement (and for Relative Complement), but those don't existYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning 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 `{{Improve}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

- $X = \paren {S \times D} \setminus \paren {U \times D} = \paren {S \setminus U} \times D$

Since $S \setminus U$ is a closed set of $\tau$, the result follows.

$\blacksquare$