CLASS NOTES MATH 527 (SPRING 2011) WEEK 3 1. Mon, Jan. 31
... weak Hausdorff space is compactly generated if a subset C ⊆ X is closed if (and only if) for every continuous map g : K −→ X with K compact, the subset g −1 (C) is closed in K. Any time from now on that we talk about spaces, we really mean compactly generated weak Hausdorff spaces. There are a coupl ...
... weak Hausdorff space is compactly generated if a subset C ⊆ X is closed if (and only if) for every continuous map g : K −→ X with K compact, the subset g −1 (C) is closed in K. Any time from now on that we talk about spaces, we really mean compactly generated weak Hausdorff spaces. There are a coupl ...
Combinatorial Equivalence Versus Topological Equivalence
... (AppO)Let U c Rn be an open set and K c U a closed piecewise linear subcomplex. Let f: U -+ R" be a topological homeomorphism which is piecewise linear on K. Let s(x) > 0 be a continuous function on U. Then there is a piecewise linear homeomorphism g: U -+ R" which agrees with f on the subcomplex K, ...
... (AppO)Let U c Rn be an open set and K c U a closed piecewise linear subcomplex. Let f: U -+ R" be a topological homeomorphism which is piecewise linear on K. Let s(x) > 0 be a continuous function on U. Then there is a piecewise linear homeomorphism g: U -+ R" which agrees with f on the subcomplex K, ...
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... being open/closed/measurable implies the same property for f (S). In fact, f (X) = Image(f ) need not even be measurable. One of the few things that can be said, however, is that f (S) is compact whenever S is compact. Analytic sets are defined in order to be stable under direct images, and their th ...
... being open/closed/measurable implies the same property for f (S). In fact, f (X) = Image(f ) need not even be measurable. One of the few things that can be said, however, is that f (S) is compact whenever S is compact. Analytic sets are defined in order to be stable under direct images, and their th ...
Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below. In this case, C is called a covering space and X the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X is a ""sufficiently good"" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the conjugacy classes of subgroups of the fundamental group of X.