Notes

Description: Definitions of Compact

Content:

Definitions of Compact.

There are several definitions of ``compact'' for topological spaces which are equivalent in \(ZFC\), but not in \(ZF^0\).  In this note we describe what is known about the relationships between these definitions in \(ZF^0\). (Also, see Note 10.)

Definition: Assume \((X,T)\) is a topological space.

  1. \((X,T)\) is Heine-Borel compact (or H-B compact or compact) if every open cover has a finite subcover. (It is shown in Herrlich [1996a] that \((X,T)\) is H-B compact if and only if every filter has an accumulation point.)
  2. \((X,T)\) is Alexandroff-Urysohn compact (or A-U compact) if every infinite subset \(E\) of \(X\) has a complete accumulation point, (i.e,, a  point \(x_o\in X\) such that for every neighborhood \(U\) of \(x_0\), \(|E\cap U|=|E|\)).
  3. \((X,T)\) is subbase compact (or S-B compact) if there is a subbase \(S\) for \(T\) such that every open cover by elements of \(S\) has a finite subcover.
  4. \((X,T)\) is Bourbaki compact (or B compact) if every ultrafilter converges in \(X\). (An (ultra)filter \(\cal F\) converges to a point \(x\) in \(X\) if ever neighborhood of \(x\) is in \(\cal F\).)
  5. \((X,T)\) is linearly compact (or L compact) if every nest of non-empty closed sets has a non-empty intersection.
  6. \((X,T)\) is sequentially compact (or SEQ compact) if every sequence has a convergent subsequence.
  7. \((X,T)\) is Weierstrass compact (or W compact) if every infinite subset has an accumulation point.

In Howard [1990] it is shown that in \(\cal N1\) the set of atoms with the discrete topology is linearly compact but not subbase compact.  It is also shown that in \(\cal N56\), the topological space \(X\) described in Howard [1990] is A-U compact but not S-B compact.  (The model \(\cal N56\) is \(\cal M2\) in Howard [1990].)

In Howard [1990] and Herrlich [1996a] it is shown that, for a given topological space \(X\) the implications in the following diagram hold. This table summarizes the notation in the big diagram below:

Notation Means...
H-B Heine-Borel compact
A-U Alexandroff-Urysohn compact
\(A\rightarrow B\) \(A\) implies \(B\)
\(A\not\rightarrow B\) \(A\) does not imply \(B\)
\(A\Rightarrow B\) \(A\) implies \(B\), but \(B\) does not imply \(A\)
\(A \not\Leftrightarrow B\) \(A\) does not imply \(B\) and \(B\) does not imply \(A\)
\(A\Leftrightarrow B\) \(A\) implies \(B\) and \(B\) implies \(A\)

\[ \begin{matrix} \boxed{\text{B}} & \Longleftarrow & \boxed{\text{H-B}} & \Longleftrightarrow & \boxed{\text{H-B}} & \kern5pt\not\kern-5pt\Longleftrightarrow & \boxed{\text{A-U}} \\ & & \big\Downarrow & & \big\Downarrow & &\kern8.5pt \diagup \kern-8.5pt\big\Updownarrow\\ \boxed{\text{A-U}} & \kern5pt\not\kern-5pt\Longleftrightarrow & \boxed{\text{L}} & \kern5pt\not\kern-5pt\longrightarrow & \boxed{\text{S-B}} & \Longleftrightarrow & \boxed{\text{S-B}} \end{matrix} \]

Finally we mention a result from Herrlich [1996a]:  The assertion that products of B compact spaces are B compact is equivalent to the statement ``AC (Form 1) or there does not exist a non-principal ultrafilter (the negation of Form 206)''.

Howard-Rubin number: 6

Type: Definitions

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