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On transitive topological group actions. (English) Zbl 1202.54028
Suppose that \(X\) is a Hausdorff space and a topological group \(G\) acts continuously and transitively on \(X\). Fix a point \(x\in X\) and let \(G_x=\{g\in G: gx=x\}\) be the stabilizer of \(x\). Observe that, for any \(h\in G\) we have \(gx=hx\) for any \(g\in hG_x\) so, letting \(u_x(hG_x)=hx\), we obtain a bijection \(u_x:G/G_x\to X\). The main theorem of the paper states that if the quotient space \(G/G_x\) has a dense Čech-complete subspace and for any non-empty open subset \(U\) of \(G/G_x\), the interior of the set \(\overline{u_x(U)}\) is non-empty, then \(u_x\) is a homeomorphism. Several applications of this result are given. It is shown, among other things, that the classical open mapping theorem of functional analysis can be easily deduced from it.
Another application is the uniform generalized Schönflies theorem which states that for any locally flat embedding \(g\) of the \((n-1)\)-dimensional sphere \(S^{n-1}\) into the \(n\)-dimensional sphere \(S^n\), if \(\varepsilon>0\) then there exists a number \(\delta>0\) such that for any locally flat embedding \(f:S^{n-1}\to S^n\) which is \(\delta\)-close to \(g\), there exists an embedding \(e:S^n\to S^n\) such that \(e\) is \(\varepsilon\)-close to the identity and \(e\circ g=f\).

54H15 Transformation groups and semigroups (topological aspects)
54H11 Topological groups (topological aspects)
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