# Tagged Questions

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### Does the converse of Tychonoff's theorem hinge on the axiom of choice?

Tychonoff's theorem:$\phantom{---}$ If $A$ is a non-empty index set and $X_{\alpha}$ is a non-empty compact topological space for every $\alpha\in A$, then $X\equiv\times_{\alpha\in A} X_{\alpha}$ is ...
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### Clarifications on proof of compactness theorem

I've been reading through the following proof of compactness theorem: http://www.princeton.edu/~hhalvors/teaching/phi312_s2013/compactness.pdf One thing that struck me is that this proof seems to ...
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### Closed subsets of $\beta \mathbb R$

Definitions. Suppose $X$ is a topological space. $w(X)=\min\{|\mathcal B|:\mathcal B$ is a base for $X\}+\omega$ $e(X)=\sup\{|D|:D\subseteq X$ is closed and discrete$\}+\omega$ $K(X)$ is the ...
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Formulas for sines or cosines of sums superficially appear to have a certain symmetry, specifically it looks as if sine and cosine play something like symmetrical roles: \begin{align} & ... 1answer 178 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. 1answer 122 views ### Is \{0,1\}^{\omega_1} sequentially compact? It is claimed that an uncountable product of [0,1] is not sequentially compact, e.g. in Wikipedia (I think replacing [0,1] by \{0,1\} doesn't make much difference). However, the constructions I ... 1answer 116 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. ... 1answer 101 views ### Show that X is separable. I'm working through Kunen's Set Theory and I'm not sure how to proceed on part of one exercise. Let X be compact Hausdorff and \mathbb{O}_X be the poset of nonempty open sets of X ordered by ... 0answers 77 views ### Is X pseudocompact The following example with a little modified from the handbook of set theoretic topology, Page 574: Let \kappa be any cardinal for which there exists a family \{H_\alpha: \alpha < \kappa\} ... 2answers 309 views ### Is there a “tree-like” proof of compactness theorem in the case of uncountably many variables? I like proofs using trees and König's lemma, since they are very visual. One of the applications of König's lemma you can show to students is proving compactness theorem for propositional calculus, ... 1answer 101 views ### Is this space countably compact Let X be a Tychonoff countably compact space and A is a subapce of X such that for any countable B \subset A we have \overline{B} \subset A. My question is this: Is this subspace A ... 1answer 109 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 ... 1answer 117 views ### A question on linearly Lindelöf space A space X is linearly lindelöf if for every open cover of X, linearly ordered by the subset relation, has a countable subcover. Question is this: How could we see that X is linearly lindelöf ... 3answers 314 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 ... 1answer 281 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 ...
I'm reading a book on ultrafilters and it tells me that showing addition on ultrafilters over the naturals (addition on the set of ultrafilters '$\beta \mathbb{N}$', defined in the standard way) is ...
### Set of ultrafilters $\beta \mathbb{N} - \mathbb{N}$ is not separable
I want to show the set of ultrafilters $\beta \mathbb{N} - \mathbb{N}$ (where $\beta \mathbb{N}$ is all ultrafilters on $\mathbb{N}$) is not separable. I know we can take as a base of \$\beta ...