# Category:Closed Extension Topology

Jump to navigation
Jump to search

This category contains results about closed extension topologies.

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 $T = \struct {S, \tau}$.

## Subcategories

This category has the following 3 subcategories, out of 3 total.

### C

### L

### P

## Pages in category "Closed Extension Topology"

The following 11 pages are in this category, out of 11 total.

### C

- Closed Extension Space is Irreducible
- Closed Extension Topology is not Hausdorff
- Closed Extension Topology is not T1
- Closed Extension Topology is not T3
- Closed Extension Topology is Topology
- Closed Sets of Closed Extension Topology
- Closure of Open Set of Closed Extension Space
- Condition for Closed Extension Space to be T0 Space
- Condition for Closed Extension Space to be T4 Space
- Condition for Closed Extension Space to be T5 Space