# Category:Countable Complement Topology

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This category contains results about Countable Complement Topology.

Definitions specific to this category can be found in Definitions/Countable Complement Topology.

Let $S$ be an infinite set whose cardinality is usually taken to be uncountable.

Let $\tau$ be the set of subsets of $S$ defined as:

- $H \in \tau \iff \complement_S \left({H}\right)$ is countable, or $H = \varnothing$

where $\complement_S \left({H}\right)$ denotes the complement of $H$ relative to $S$.

In this definition, countable is used in its meaning that includes finite.

Then $\tau$ is the **countable complement topology on $S$**, and the topological space $T = \left({S, \tau}\right)$ is a **countable complement space**.

## Pages in category "Countable Complement Topology"

The following 25 pages are in this category, out of 25 total.

### C

- Compact Sets in Countable Complement Space
- Countable Complement Space is Connected
- Countable Complement Space is Irreducible
- Countable Complement Space is Lindelöf
- Countable Complement Space is Locally Connected
- Countable Complement Space is not Countably Compact
- Countable Complement Space is not Countably Metacompact
- Countable Complement Space is not First-Countable
- Countable Complement Space is not Separable
- Countable Complement Space is not Sigma-Compact
- Countable Complement Space is not T2
- Countable Complement Space is not T3, T4 or T5
- Countable Complement Space is not Weakly Countably Compact
- Countable Complement Space is Pseudocompact
- Countable Complement Space is T1
- Countable Complement Space Satisfies Countable Chain Condition
- Countable Complement Topology is Expansion of Finite Complement Topology
- Countable Complement Topology is Topology