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32 votes
3 answers
6k views

Is "compact implies sequentially compact" consistent with ZF?

Over at the nForum, we've been discussing sequential compactness. The discussion led me to realise that I naively assumed that nets were simply Big Sequences, and that I could make a reasonable guess ...
Andrew Stacey's user avatar
15 votes
2 answers
904 views

Does there exist a supercompactness theorem?

Large cardinals such as weakly compact cardinals, measurable cardinals, strongly compact cardinals, and extendible cardinals all can be characterized in terms of a certain compactness theorem of ...
Joseph Van Name's user avatar
11 votes
3 answers
890 views

Structure theorems for compact sets of rationals

Everyone knows the Heine-Borel theorem characterizing compact subsets of Euclidean space. For any $n \in \mathbb N$ a set $A \subseteq \mathbb R^n$ is compact just in case it is closed and bounded (in ...
Corey Bacal Switzer's user avatar
11 votes
0 answers
273 views

A ZFC-example of a countably compact paratopological group which is not a topological group

Problem. Does there exist a ZFC-example of a countably compact Hausdorff paratopological group which is not a topological group? (The problem posed 27 May 2015 by Alexander Ravsky on page 9 of Volume ...
Lviv Scottish Book's user avatar
1 vote
3 answers
688 views

How to show the cardinality of nonisometric compact metric spaces is the continuum

It is asserted in A Course in Metric Geometry by Burago, Burago, Ivanov that there can be no more than continuum of mutually nonisometric compact spaces How is this proven? Its clear that there ...
Otis Chodosh's user avatar
  • 7,197
1 vote
1 answer
124 views

Is bounded-$\mathcal L_{\omega_1,\omega_1}$ significantly different from $\mathcal L_{\omega_1,\omega_1}$?

Take the language $\mathcal L(=,\in)_{\omega_1,\omega_1}$, if we restrict infinite quantification strings, i.e. of the forms $``(\forall v_i)_{i \in \omega}"; ``(\exists v_i)_{i \in \omega}"$, to ...
Zuhair Al-Johar's user avatar