Tagged Questions

5
votes
0answers
89 views

Is dependent choice necessary to prove every perfect compact Hausdorff space is uncountable?

The answer to Cardinality of a locally compact space without isolated point shows that AC is required to show that if $X$ is a compact Hausdorff space with no isolated points then $|X| \ge ...
3
votes
1answer
70 views

Tychonoff Theorem and the axiom of choice

How to show that The Tychonoff Theorem and the axiom of choice are equivalent? Here I want to collect ways to prove it. Thanks for your help.
3
votes
1answer
84 views

Tychonoff Theorem in the Realm of $\neg AC$

It's widely know that the Tychonoff Theorem is equivalent to the Axiom of Choice; thus, assuming the negation of the axiom of choice, I'd like to know if there is a canonical example of a collection ...
11
votes
1answer
180 views

Cardinality of a locally compact space without isolated point

I am interested in the following result: Theorem: A locally compact and Hausdorff topological space $X$ without isolated point has at least the cardinality $\mathfrak{c}$. To prove it, one can ...
3
votes
1answer
67 views

Is compact metric space separable in ZF?

Reference; http://www.samos.aegean.gr/math/kker/papers/CompactMetric.pdf The paper says "Compact metric space is separable" is unprovable in ZF$^0$( That is, ZF without axiom of regularity). And i ...
2
votes
2answers
166 views

Is the negation of Gödel's completeness theorem consistent with $ZF$ without AC?

The proof of compactness and completeness of $\mathscr{FOL}$ (with Hilbert system) used Zorn's lemma. And Zorn's lemma is equivalent to the Axiom of choice in $ZF$. So my question is can they be ...
3
votes
3answers
236 views

If a metric space has the limit point property, is it separable? (ZF + AC$_\omega$)

If a metric space has the limit point property, is it separable? (ZF + AC$_\omega$) I'm struggling with this problem for a week. I'm talking about this in Metric space. Here's the part of ...
4
votes
1answer
147 views

Is a countable product of compact intervals in $\mathbf R$ compact (without using the AC)?

Let $\{I_n=[a_n,b_n]\}_{n\in\mathbf N}$ be a countable collection of closed, bounded intervals in $\mathbf R$. Is the infinite Cartesian product $$\prod_{n=1}^\infty I_n$$ compact without using the ...
1
vote
2answers
196 views

Heine-Borel Theorem ($\mathbb{R}^k$) (in ZF)

Heine-Borel Theorem; If $E \subset \mathbb{R}^k$, then $E$ is compact iff $E$ is closed and bounded. I have proved 'closed and bounded⇒compact' and 'compact⇒bounded'. (There exists $r\in \mathbb{R}$ ...
3
votes
3answers
236 views

Axiom of choice and compactness.

I was answering a question recently that dealt with compactness in general topological spaces, and how compactness fails to be equivalent with sequential compactness unlike in metric spaces. The only ...