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Description: Definitions of Compact

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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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