Find a cardinality of a particular topology.

Question

If X is an infinite set and a Topology T is defined over it as:

$T = \{ A \subset X \ | \ A = \emptyset \ \lor A^C - finite \}$

How is it possible to find the $card(T)$ above?

I guess one would have to find a bijection to some other set, the cardinality of which is easier to find. But which one?

The cardinality of $T$ varies, depending on the cardinality of $X$. — GEdgar, Jan 11 '20 at 22:44

Henno Brandsma · Accepted Answer · 2020-01-11T22:46:31.290

3

If $X$ is infinite, then the set of finite subsets of $X$ has the same cardinality as $X$ (see e.g. here). So the set of complements of finite sets (which is in bijection with the set of finite sets, trivially) also has cardinality $\text{card}(X)$. Adding the singleton $\{\emptyset\}$ to an already infinite set doesn't change that, so the answer is just $|X|$ or $\text{card}(X)$.

edited Jan 11 '20 at 22:46

answered Jan 11 '20 at 22:43

Henno Brandsma

250,824

If $X$ is finite, then $T=P(X)$ is just the parts of $X$ ;) – N. S. Jan 11 '20 at 22:44
1

@N.S. The question states $X$ is infinite. – Henno Brandsma Jan 11 '20 at 22:45

Find a cardinality of a particular topology.

1 Answers1