Definition:Arens-Fort Space
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Definition
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.
Also see
- Results about the Arens-Fort space can be found here.
Source of Name
This entry was named for Richard Friederich Arens and Marion Kirkland Fort, Jr.
Historical Note
The Arens-Fort space was created by Richard Friederich Arens as an adaption of a Fort Space.
Sources
- 1950: Richard Friederich Arens: Note on convergence in topology (Math. Mag. Vol. 23: pp. 229 – 234) www.jstor.org/stable/3028991
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $26$. Arens-Fort Space