-5
votes
1answer
503 views

Proof that a product of two quasi-compact spaces is quasi-compact without Axiom of Choice [closed]

A topological space is called quasi-compact if every open cover of it has a finite subcover. Let $X, Y$ be quasi-compact spaces, $Z = X\times Y$. The usual proof that $Z$ is quasi-compact uses a ...
1
vote
0answers
37 views

Amount to choice necessary to prove instances of Tychonoff theorem

Let $I$ be a fixed nonempty set. I would like to know how much choice is necessary in order to prove that the product of any $I$-indexed family of compact topological spaces is compact (under the ...
8
votes
2answers
84 views

How should one think about results that depend on AC?

I just encountered this: "(Theorem of A. H. Stone) Every metric space is paracompact... Existing proofs of this require the axiom of choice... It has been shown that neither ZF theory nor ZF ...
5
votes
1answer
107 views

Compact metric spaces is second countable and axiom of countable choice

Why we need axiom of countable choice to prove following theorem: every compact metric spaces is second countable? In which step it's "hidden"? Thank you for any help.
3
votes
1answer
103 views

Is there anything wrong with this use of the axiom of choice?

I have a proof of the following theorem: Let A be a finite set and X be a perfectly normal topological space, and let $\{(A,\succsim_x): x\in X\}$ be a family of binary relations on $A$ satisfying ...
6
votes
1answer
142 views

Can we prove that every ordered space is normal without choice?

In ZFC, every linear ordered space respect to the order topology is completely normal. I saw the this proof and proof of this statement in the book "Counterexamples of topology" (Example 39). But as I ...
0
votes
0answers
28 views

Is it possible to prove product of basis is a basis for the box topology without AC?

Let $\{(X_i,\tau_i)\}$ be a collection of topological spaces and $\mathscr{B}_i$ be a basis for $\tau_i$. Let $\mathbb{B}=\{\prod p_i : p\in \prod \mathscr{B}_i\}$. Then is $\mathbb{B}$ a basis for ...
4
votes
1answer
105 views

Question regarding disjoint unions, sequential compactness, and Dedekind-finiteness

I have proved the following two results: $[\mathsf{ZF}]$ The disjoint union of a Dedekind-finite family of sequentially compact topological spaces is again sequentially compact. ...
1
vote
1answer
65 views

Showing that if every open subspace is Lindelöf, then the space is hereditarily Lindelöf.

Background: A topological space $X$ is said to be Lindelöf if for every cover $\mathcal O$ of $X$ by open subsets of $X$, there is some $\mathcal C\subseteq\mathcal O$ such that $\mathcal C$ is a ...
1
vote
1answer
125 views

Using Zorn's Lemma-Topology

Question: A subset $W$ of the set $Z$ of integers is said to be closed under addition if given any elements $w$ and $w'$ of $W$, $w+w'\in W$. Prove that there is a maximal subset of $Z$ which is ...
0
votes
1answer
54 views

Tychonoff's theorem and axiom of choice in Hausdorff spaces

Does anyone know if axiom of choice is nessesary in proving Tychonoff's theorem in a Hausdorff space? Thanks!
5
votes
1answer
109 views

Relationships between a few strong separation axioms without Choice

Disambiguation: Let $\langle X,\tau\rangle$ be a topological space. I will say that two subsets $A,B$ of $X$ are separated if they are disjoint from each other's closures (their closures needn't be ...
2
votes
1answer
29 views

How can one tell if a sequence is well-defined; is the axiom of choice needed?

This question is in the context of the following exercise: Let $X$ be a first countable topological space, let $A \subseteq X$, and let $x \in X$ with $x \in \overline{A}$. Then there exists a ...
2
votes
1answer
97 views

Cluster point of a sequence $\{x_n\}$ is the limit of some subsequence - Axiom of Choice? [duplicate]

In a metric space, a cluster point of a sequence $\{x_n\}$ is the limit of some subsequence. The only proof that I know works like this: Construct a sequence $\delta _k \to 0$. For each $\delta _k$ ...
6
votes
0answers
201 views

A few standard results (on metrizability and relative separation strength) without choice?

I've been going back over some results from Munkres's Topology, and I'm curious about some things.... I know that Choice principles have some connection to the separation axioms (in ZF, at ...
5
votes
0answers
117 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 ...
2
votes
1answer
107 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
118 views

well-ordering principle

I'm confused by the well-ordering principle . The proof is clear but I don't have any idea why is it true . It says that every non-empty set can be well-ordered but $C^{1}$ is a non-empty set but ...
6
votes
3answers
204 views

A question about a proof in one of Sierpiński's papers

The following is a question about Sierpiński's paper "Une démonstration du théorème sur la structure des ensembles de points", (link): We call a set dense-in-itself if it does not contain any ...
4
votes
2answers
379 views

Second-countable implies separable/Axiom countable choice

Let $(X,\mathscr T)$ be a topological space, and $(B_n)_{n\ge1}$ a countable basis for X. Under this assumptions, X is separable. The proof of this assertion is as follows: We can assume without ...
3
votes
1answer
101 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 ...
3
votes
1answer
131 views

What is Alexander subbase theorem equivalent to?

I read that Alexander subbase theorem is equivalent to the Boolean prime ideal theorem which is weaker than AC. But AC implies Alexander subbase theorem and subbase theorem implies Tychonoff, see ...
14
votes
1answer
299 views

Cardinality of a locally compact Hausdorff space without isolated points

I am interested in the following result: Theorem. A locally compact Hausdorff topological space $X$ without isolated points has at least cardinality $\mathfrak{c}$. To prove it, one can find two ...
6
votes
0answers
140 views

What are 'weak' forms of Urysohn's lemma, which do not require choice?

Reference: http://web.mat.bham.ac.uk/C.Good/research/pdfs/horror.pdf It is well known that the original proof of Urysohn's lemma uses a choice principle. (DC) What are weak forms of the Urysohn's ...
4
votes
1answer
246 views

Totally bounded, sequentially compact, complete, bounded, closed, equicontinuous $\Rightarrow$ compact?

Related; When $K$ is compact, if $S\subset C_b(K)$ is closed,bounded and equicontinuous, then $S$ is compact? (ZF) I just edited my whole question since i think it was a bit messy. Here is my ...
1
vote
1answer
84 views

Explicit choice functions for finite sets in topological spaces

When dealing with finite nonempty sets of real or natural numbers it is always possible to define a explicit choice function, that choose one (arbitrary, but well defined) element out of that set: ...
4
votes
1answer
99 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 ...
7
votes
2answers
216 views

Relationships between AC, Ultrafilter Lemma/BPIT, Non-measurable sets

How is it possible to reconcile the following... In 1970, Solovay constructed Solovay's model, which shows that it is consistent with standard set theory, excluding uncountable choice, that all ...
0
votes
1answer
70 views

Connected set in $\mathbb{R}$ and constructing a sequence to a limit point

Let $C$ be a connected set in $\mathbb{R}$. Let $p$ be a limit point of $C$. Let $f:C\rightarrow \mathbb{R}$ be a function. Suppose that; $\exists \epsilon>0$ such that [ $\forall \delta>0, ...
1
vote
1answer
73 views

Some properties of Polish space

Let $X$ be a separable complete metric space. I wonder if following properties hold in ZF. Limit Compact ⇒ Compact Does there exist a function$f$ such that $f(E)$ is closed and $f(E)\subset E$, for ...
7
votes
1answer
153 views

Constructing a choice function in a complete & separable metric space

Let $X$ be a complete & separable metric space. Let $\{E_i\}_{i\in I}$ be a collection of closed and nonempty sets in $X$. If $X$ is just a complete metric space, it seems not possible to ...
0
votes
1answer
66 views

Analysis Proof without Axiom of Choice

Lemma: Given any function $f:M\to N$ where $M$ and $N$ are both metric spaces, $\lim_{x\to a}f(x)$ converges to $L$ only if given any function $\gamma:\mathbb{I}\to M$ (where $\mathbb{I}$ is the unit ...
0
votes
1answer
129 views

Discontinuous Functions on the Real Line

I want to prove that given any function $g:\mathbb{Z} \to \mathbb{R}$ there exists a function $f:\mathbb{R}\to\mathbb{R}$ such that its restriction to the integers is equal to g and such that it not ...
3
votes
3answers
731 views

(ZF) Prove 'the set of all subsequential limits of a sequence in a metric space is closed.

Let $X$ be a metric space. Let $\{p_n\}$ be a sequence in $X$. Let $E$ be a set of all subsequential limits of $\{p_n\}$. How do i prove that $E$ is closed in ZF? Is there a well-ordering of ...
1
vote
1answer
146 views

(ZF) Equivalent statements to Baire Category Theorem

So far, I have proved following two for a polish space $X$; 1.If $\{F_n\}$ is a family of closed subset of $X$, where $X=\bigcup_{n\in \omega} F_n$, then at least one $F_n$ has a nonempty inteior. ...
4
votes
4answers
281 views

(ZF) If $\mathbb{R}^k$ is a countable union of closed sets, then at least one has a nonempty interior

Since the specific space $\mathbb{R}^k$ is given, this might be provable in ZF. Let $\{F_n\}_{n\in \omega}$ be a family of closed subset of $\mathbb{R}^k$, of which the union is $\mathbb{R}^k$. ...
3
votes
3answers
288 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 ...
1
vote
1answer
101 views

(ZF) Dedekind infinite + Limit Point Compact ⇒ Separable

If every infinite subset has a limit point in a metric space $X$, then $X$ is separable (in ZF) Yesterday, i posted this question and got an answer that 'Limit Point Compact⇒Separable' is unprovable ...
5
votes
2answers
340 views

If every infinite subset has a limit point in a metric space $X$, then $X$ is separable (in ZF)

I can prove this in ZFC, but don't know how to prove this in ZF. Following is the argument of this in ZFC. Fix $0<r\in \mathbb{R}$ and $x_0\in X$. Let $A_i = \{x\in X\mid d(x,x_j)\ge r,\, ...
0
votes
1answer
122 views

Showing the bijection between countable dense subset of a metric space $E×\mathbb{Q}→$neighborhoods of $E$ with rational radius

Let $X$ be a separable metric space. Let $E$ be a countable dense subset of $X$. Let $p_i$ enumerate $E$ and $q_j$ enumerate $\mathbb{Q}$. Let $G=\{N(p_i,q_j) \subset X | i,j\in \omega\}$ Then $G$ is ...
4
votes
1answer
250 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 ...
0
votes
1answer
277 views

(ZF) Every nonempty perfect set in $\mathbb{R}^k$ is uncountable.

This is the part of proof in Rudin PMA p.41 Let $P(\subset \mathbb{R})$ be a perfect set. Since $P$ has limit points, $P$must be infinite. Suppose that $P$ is countable. Then, we can denote the ...
1
vote
2answers
248 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 $r\in \mathbb{R}$ ...
0
votes
0answers
409 views

Every k-cell is compact (in ZF)

This is the part of proof on my book. Let $I$ be a k-cell. Then for every $x\in I$ and $j\in k$, $a_j ≦ x(j) ≦ b_j$ for some $a_j, b_j \in \mathbb{R}$. Let {$G_{\alpha}$} be an open cover of I and ...
0
votes
1answer
84 views

Open relative and choice

I'm using ZF as my axiom system. Let $X$ be a metric space and $K\subset Y \subset X$. Suppose $K$ is compact relative to $X$. Let $\{V_a\mid a\in I\}$ be a family of open sets relative to $Y$ such ...
4
votes
3answers
312 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 ...
2
votes
3answers
275 views

Order topology and axiom of choice

For a linearly (totally) ordered set $A$, one can define its order topology: that is the smallest topology containing the set B of all the intervals of the form $\{x\mid x < a\}, \{x\mid x > ...
7
votes
2answers
728 views

Maximal ideals in $C(X)$ and Axiom of Choice

The following result are true if we assume full axiom of choice: A. If $X$ is a compact Hausdorff space, then every maximal ideal of the ring $C(X)$ has the form $A_p=\{f\in C(X); f(p)=0\}$. B. If ...
10
votes
2answers
365 views

Does proving (second countable) $\Rightarrow$ (Lindelöf) require the axiom of choice?

Background: This question came up in my homework (but was not a homework problem). The problem was proving one direction of the Heine-Borel theorem. As with all proofs of compactness, one begins ...
4
votes
2answers
416 views

A concrete example of a choice function

I'm trying to understand a bit what lies behind the Axiom of Choice, and I was wondering, are there concrete examples of a choice function on the Borel set? The Borel set seems nice enough for a ...