# Category:Open Extension Topology

Jump to navigation
Jump to search

This category contains results about **open extension topologies**.

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

Let $p$ be a new element which is not in $S$.

Let $S^*_p = S \cup \set p$.

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

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

That is, $\tau^*_{\bar p}$ is the set of all sets formed by taking all the open sets of $\tau$ and adding to them the set $S^*_p$.

Then:

- $\tau^*_{\bar p}$ is the open extension topology of $\tau$

and:

- $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ is the open extension space of $T = \struct {S, \tau}$.

## Subcategories

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

## Pages in category "Open Extension Topology"

The following 18 pages are in this category, out of 18 total.

### C

### O

- Open Extension of Double Pointed Countable Complement Topology is T4 Space
- Open Extension Space is Compact
- Open Extension Space is Connected
- Open Extension Space is Path-Connected
- Open Extension Space is Ultraconnected
- Open Extension Topology is not Perfectly T4
- Open Extension Topology is not T1
- Open Extension Topology is not T3
- Open Extension Topology is T4
- Open Extension Topology is Topology