12
votes
Accepted
Pointed versus unpointed maps into a topological monoid
As is implicitly pointed out in the comments, you really want to assume that $X$ ($=M$) is path connected. And then your analysis is fine. Note that $M$ will then wish to be equivalent to $\Omega BM$...
- 11.1k
10
votes
Accepted
Group completion of topological monoids
A suitable counterexample can be constructed as follows.
Let $(e_n)_{n\in\omega}$ be the standard orthonormal basis of the Hilbert space $\ell_2$. For every $n\in\mathbb N$ consider the linear hull $...
- 41.8k
10
votes
Accepted
Stone–Čech compactification as a semigroup
Corollary 4.33 of Hindman and Strauss's book on Algebra in the Stone Cech Compactification says that if $S$ is an infinite cancellative (discrete) semigroup, then the nonprincipal ultrafilters in $\...
- 38.6k
9
votes
Accepted
Why is choice needed in Ellis' Lemma?
The issue is that $R$ could be empty. When you apply Zorn's lemma, a nested intersection of non-empty compact sets is non-empty, guaranteeing minimal non-empty elements.
- 2,071
4
votes
Accepted
Continuity of Moore-Penrose generalized inversion
The answer is indicated by the geometric description of the Moore-Penrose generalized inverse, which is shown here. The observation is that the restriction of a nonzero matrix to the orthogonal ...
- 27.5k
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