By A. V. Arhangel’skii, J. van Mill (auth.), K.P. Hart, J. van Mill, P. Simon (eds.)
The publication provides surveys describing contemporary advancements in lots of the fundamental subfields of normal Topology, and its functions to Algebra and research over the last decade, following the former variants (North Holland, 1992 and 2002). The booklet was once ready in reference to the Prague Topological Symposium, held in 2011. over the last 10 years the point of interest normally Topology replaced and for that reason the choice of subject matters differs from that selected in 2002. the subsequent components skilled major advancements: Fractals, Coarse Geometry/Topology, size idea, Set Theoretic Topology and Dynamical Systems.
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The booklet offers surveys describing fresh advancements in many of the basic subfields of common Topology, and its functions to Algebra and research over the past decade, following the former variations (North Holland, 1992 and 2002). The publication was once ready in reference to the Prague Topological Symposium, held in 2011.
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2 () Any space X is pc-twistable at any chain-point. Proof Let e be a chain-point in X , and γ be a strong chain at e. Take any x, y ∈ X . Put x y = y if there exists V ∈ γ such that x ∈ V and y ∈ / V . Otherwise, put x y = x. In particular, it follows that ey = y, for each y ∈ X , and xe = x, for each x ∈ X . It cannot occur that, for some V, U ∈ γ and x, y ∈ X , x ∈ V , y ∈ / V , y ∈ U, and x ∈ / U , since γ is a chain. Therefore, the definition of multiplication is correct. Let us check that the binary operation so defined is a point-continuous twister on X at e.
1 that the G τ -tightness of X λ at least at one point does not exceed τ . Hence, the G τ -tightness of X λ is not greater than τ at all points, since the space X λ is homogeneous. The natural projection of X λ onto X is open and continuous. 2, the G τ -tightness of X also does not exceed τ . The set of G τ -points in X is not empty. 6 that X is τ -twistable at some point. 4, X λ is τ -twistable at some point. Since X λ is homogeneous, the space X λ is τ -twistable. 3 that X is τ -twistable. 5.
Subsequently, Hernández-Gutiérrez and Hrušák  showed that for every non-meager P-filter F on ω, both F and F ω are CDH. The topological sum of the 1-sphere S1 and S2 is an example of a CDH-space that is not homogeneous. R. Bennett  proved in 1972 that a connected firstcountable CDH-space is homogeneous. ) Hence for connected metrizable spaces, countable dense homogeneity can be thought of as a strong form of homogeneity. Topological Homogeneity 47 After 1972, the interest in CDH-spaces was kept alive mainly by Fitzpatrick.
Recent Progress in General Topology III by A. V. Arhangel’skii, J. van Mill (auth.), K.P. Hart, J. van Mill, P. Simon (eds.)