# Category:Arens-Fort Space

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This category contains results about the Arens-Fort space.

Let $S$ be the set $\Z_{\ge 0} \times \Z_{\ge 0}$ be the Cartesian product of the set of positive integers $\Z_{\ge 0}$:

- $S = \set {0, 1, 2, \ldots} \times \set {0, 1, 2, \ldots}$

Let $\tau \subseteq \powerset S$ be a subset of the power set of $S$ such that:

- $(1): \quad \forall H \subseteq S: \tuple {0, 0} \notin H \implies H \in \tau$

- $(2): \quad H \subseteq S: \tuple {0, 0} \in H$ and, for all but a finite number of $m \in \Z_{\ge 0}$, the sets $S_m$ defined as:
- $S_m = \set {n: \tuple {m, n} \notin H}$

- are finite.

That is, $H$ is allowed to be in $\tau$ if, considering $S = \Z_{\ge 0} \times \Z_{\ge 0}$ as the lattice points of the first quadrant of a Cartesian plane:

Either:

- $H$ does not contain $\tuple {0, 0}$

- $H$ contains $\tuple {0, 0}$, and only a finite number of the columns of $S$ are allowed to omit an infinite number of points in $H$.

Then $\tau$ is the **Arens-Fort topology** on $S = \Z_{\ge 0} \times \Z_{\ge 0}$, and the topological space $T = \struct {S, \tau}$ is the **Arens-Fort space**.

## Subcategories

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

### A

## Pages in category "Arens-Fort Space"

The following 26 pages are in this category, out of 26 total.

### A

- Arens-Fort Space is Completely Hausdorff
- Arens-Fort Space is Completely Normal
- Arens-Fort Space is Countable
- Arens-Fort Space is Expansion of Countable Fort Space
- Arens-Fort Space is Lindelöf
- Arens-Fort Space is Non-Meager
- Arens-Fort Space is not Compact
- Arens-Fort Space is not Connected
- Arens-Fort Space is not Countably Compact
- Arens-Fort Space is not Extremally Disconnected
- Arens-Fort Space is not First-Countable
- Arens-Fort Space is not Locally Connected
- Arens-Fort Space is not Weakly Locally Compact
- Arens-Fort Space is Scattered
- Arens-Fort Space is Separable
- Arens-Fort Space is Sigma-Compact
- Arens-Fort Space is T1
- Arens-Fort Space is T5
- Arens-Fort Space is Totally Separated
- Arens-Fort Space is Zero Dimensional
- Arens-Fort Topology is Topology