# Category:Compact Complement Topology

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This category contains results about Compact Complement Topology.

Let $T = \struct {\R, \tau}$ be the real number line with the usual (Euclidean) topology.

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

- $\tau^* = \leftset {S \subseteq \R: S = \O \text { or } \relcomp \R S}$ is compact in $\rightset {\struct {\R, \tau} }$

where $\relcomp \R S$ denotes the complement of $S$ in $\R$.

Then $\tau^*$ is the **compact complement topology** on $\R$, and $T^* = \struct {\R, \tau^*}$ is the **compact complement space** on $\R$.

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Compact Complement Topology"

The following 15 pages are in this category, out of 15 total.

### C

- Compact Complement Space is not T2, T3, T4 or T5
- Compact Complement Topology is Coarser than Euclidean Topology
- Compact Complement Topology is Compact
- Compact Complement Topology is Connected
- Compact Complement Topology is First-Countable
- Compact Complement Topology is Irreducible
- Compact Complement Topology is Locally Connected
- Compact Complement Topology is not Ultraconnected
- Compact Complement Topology is Second-Countable
- Compact Complement Topology is Separable
- Compact Complement Topology is Sequentially Compact
- Compact Complement Topology is T1
- Compact Complement Topology is Topology
- Countable Local Basis in Compact Complement Topology