# Definition:Closed Extension Topology

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## Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $p$ be a new element for $S$ such that $S^*_p := S \cup \set p$.

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

- $\tau^*_p := \set {U \cup \set p: U \in \tau} \cup \set \O$

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:

- $T^*_p := \struct {S^*_p, \tau^*_p}$ is the closed extension space of $T = \struct {S, \tau}$.

## 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

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $12$. Closed Extension Topology: $20$