By Jan J. Dijkstra

Permit M be both a topological manifold, a Hilbert dice manifold, or a Menger manifold and allow D be an arbitrary countable dense subset of M. give some thought to the topological workforce \mathcal{H}(M,D) which is composed of all autohomeomorphisms of M that map D onto itself outfitted with the compact-open topology. The authors current a whole strategy to the topological type challenge for \mathcal{H}(M,D) as follows. If M is a one-dimensional topological manifold, then they proved in an prior paper that \mathcal{H}(M,D) is homeomorphic to \mathbb{Q}^\omega, the countable energy of the distance of rational numbers. In all different situations they locate during this paper that \mathcal{H}(M,D) is homeomorphic to the famed Erds house \mathfrak E, which is composed of the vectors in Hilbert house \ell^2 with rational coordinates. They receive the second one consequence by means of constructing topological characterizations of Erds house

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Let s, t ∈ T such that |s| = |t| and s = t. Then Es ∩ Et = ∅ and hence there is a D ∈ C that separates the two sets. Thus Ys and Yt must also be disjoint. So we have that (Yr )r∈T satisﬁes the disjointness condition (5). We now verify that nski stratiﬁcation for X. It is obvious that the system satisﬁes (Xr )r∈T is a Sierpi´ condition (ii), that X∅ = X, and that Xt ⊂ Xs whenever s ≺ t. Let s ∈ T and let xσ ∈ Xs . Put k = |s| and note that xσ ∈ Ys and xσ ∈ Yσ k . By disjointness we have that σ k = s.

15), μ2 μ2 μ1 M (ψ W0 ) ≤ log = log < r + δ = t2−n . + log M (ϕ W0 ) ϕ(p1 ) μ1 ϕ(p1 ) and, analogously, log(M (ϕ W0 )/M (ψ W0 )) < t2−n . We deﬁne WU = {W0 , W1 , . . 19) Un+1 = {U ∈ Un : U ∩ A = ∅} ∪ {WU : U ∈ Un , U ∩ A = ∅}. and note that the induction hypotheses for n + 1 are satisﬁed. 20) V = {U : U ∈ Un for some n and U ∩ A = ∅}. Hypothesis (4) implies that V = C \ A. It follows from hypotheses (1) and (2) that V is a partition of C \ A. Let U be an element of V and let n be the ﬁrst integer such that U ∈ Un .

11. The cohesion concept also plays an important role in characterizing complete Erd˝os space. For instance, the proof above can easily be adapted to show that a nonempty space E is homeomorphic to Ec if and only if E is cohesive and there is a topology T on E that witnesses the almost zero-dimensionality of E such that every point in E has a neighbourhood that is compact in (E, T). This and other characterizations of Ec can be found in Dijkstra and van Mill [22]. 19. CHAPTER 6 Unknotting Lelek functions Let ϕ : X → R+ and ψ : Y → R+ be functions.