Definition:Maximal Compact Topology
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Definition
Let $X = \Z_{>0} \times \Z_{>0}$ be the set of all lattice points $\tuple {i, j}$ of the Cartesian plane where $i$ and $j$ are both (strictly) positive integers.
Let $S = X \cup \set {x, y}$ where $x$ and $y$ are two new elements of $X$.
Let $\tau$ be the topology defined on $S$ as follows:
- Each lattice point of $S$ is an open point.
- The open neighborhoods of $x$ are of the form $S \setminus A$ where $A$ is any set of lattice points with at most finitely many points on each row
- The open neighborhoods of $y$ are of the form $S \setminus B$ where $B$ is any set of lattice points selected from at most finitely many rows.
$\tau$ is referred to as the maximal compact topology.
The topological space $T = \struct {S, \tau}$ is referred to as the maximal compact space.
Also see
- Results about the maximal compact topology can be found here.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): Part $\text {II}$: Counterexamples: $99$. Maximal Compact Topology