# Definition:Closed Extension Topology

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## Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $p$ be a new element for $S$ such that $S^*_p = S \cup \left\{{p}\right\}$.

Let $\tau^*_p$ be the set defined as:

- $\tau^*_p = \left\{{U \cup \left\{{p}\right\}: U \in \tau}\right\} \cup \left\{{\varnothing}\right\}$

That is, $\tau^*_p$ is the set of all sets formed by adding $p$ to all the open sets of $\tau$ and including the empty set.

Then $\tau^*_p$ is the closed extension topology of $\tau$, and $T^*_p = \left({S^*_p, \tau^*_p}\right)$ is the closed extension space of $T = \left({S, \tau}\right)$.

## Also see

- Closed Extension Topology is Topology
- Closed Sets of Closed Extension Topology (which explains the name closed extension topology).
- Open Exension Topology

- Results about
**closed extension topologies**can be found here.

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 12: \ 20$