# Category:Compact Complement Topology

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This category contains results about Compact Complement Topology.

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

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

- $\tau^* = \left\{{S \subseteq \R: S = \varnothing \text { or } \complement_\R \left({S}\right)}\right.$ is compact in $\left.{\left({\R, \tau}\right)}\right\}$

where $\complement_\R \left({S}\right)$ denotes the complement of $S$ in $\R$.

Then $\tau^*$ is the **compact complement topology** on $\R$, and $T^* = \left({\R, \tau^*}\right)$ 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