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Results tagged with gn.general-topology
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user 165275
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
8
votes
Accepted
Homotopy type of the Hausdorff metric
In J. Andres, M. Väth, Calculation of Lefschetz and Nielsen Numbers in Hyperspaces for Fractals and Dynamical Systems, Proc. Amer. Math. Soc. 135 (2007), 479-487, it was shown (esssentially, the resul …
- 1,869
5
votes
BCT equivalent to DC
Yet another formulation of Blair's proof is in M. Väth, Topological Analysis, DeGruyter 2012.
- 1,869
2
votes
What do absolute neighborhood retracts look like?
For Euclidean neighborhood retracts, there is the nice characterization of being locally connected (and locally compact). Unfortunately, in the infinite-dimensional case, locally connectedness is nece …
- 1,869
2
votes
Under what general conditions is the set $S := \left\{\int_{X}v(x)\pi(x)\,\mathrm{d}P(x) \mi...
I guess that if $v$ is $P$-integrable then the answer is positive, and actually the set is compact.
Indeed, what you are looking for in this case is the compactness of the Aumann integral of the measu …
- 1,869
2
votes
What is your favorite proof of Tychonoff's Theorem?
Late to the party, but I think still worth mentioning as it seems to be relatively unknown:
The “simple proof” given by D, G. Wright is particularly nice, because
It uses only the definition of compa …
1
vote
Axiom of Countable Choice and meager sets
What is your definition of meager in (ZF)? Being a countable union of nowhere dense sets? In that case, the given argument about the failure of (UMM) in (ZF) is not clear: It is not obvious (to me) th …
- 1,869
1
vote
Relative compactness in topological spaces (reference request)
My guess is that even $T_3$ is already sufficient. I do not have access to the monograph Fletcher, Peter and Lindgren, William F., Quasi-uniform spaces, M. Dekker, New York, Basel 1982, in the moment, …
- 1,869
1
vote
Accepted
Continuity of Kakutani fixed points
I assume that you mean that $F$ is upper semicontinuous on the product space. Then in particular (since $X$ is compact, Hausdorff and $F$ has closed values), $F$ has a closed graph. This implies that …
- 1,869
1
vote
Accepted
Proving neighborhood of a compact product space contains a sub-neighborhood formed by taking...
Algernon's argument seems to need a special case of Tychonoff's theorem to get to the finiteness of the union. Here is an argument which avoids that.
Lemma: Let $A\subseteq X$ be compact and $B\subset …
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