Alexander's Compactness Theorem
Let $\BB$ be a sub-basis of $\tau$.
- $V$ is an open set
- $V \notin \CC$
then $\CC \cup \set V$ has a finite subcover, necessarily of the form:
- $\CC_0 \cup \set V$
Suppose $\CC \cap \BB$ covers $S$.
But $\CC$ does not have a finite subcover.
So $\CC \cap \BB$ does not cover $S$.
Let $x \in S$ that is not covered by $\CC \cap \BB$.
We have that $\CC$ covers $S$, so:
- $\exists U \in \CC: x \in U$
We have that $\BB$ is a sub-basis.
So for some $B_1, \ldots, B_n \in \BB$, we have that:
- $x \in B_1 \cap \cdots \cap B_n \subseteq U$
Since $x$ is not covered, $B_i \notin \CC$.
But then $U$ and all the $\CC_i$ cover $S$.
Hence $\CC$ has a finite subcover.
Axiom of Choice
Most mathematicians are convinced of its truth and insist that it should nowadays be generally accepted.
However, others consider its implications so counter-intuitive and nonsensical that they adopt the philosophical position that it cannot be true.
Instead, it relies on the intermediate Ultrafilter Lemma.
Also known as
Alexander's Compactness Theorem is also known as:
which can also be seen in the form the Alexander Sub-Basis Theorem, and so on.
Sub-Base and Sub-Basis can also be seen here rendered as Subbase and Subbasis.
Source of Name
This entry was named for James Waddell Alexander II.
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Global Compactness Properties
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Entry: Alexander's sub-base theorem