Lecture 8
... non-compact spaces, it is natural to ask the following question: can a given space be embedded homeomorphically in a compact space? The answer to this is always yes. In fact, there are many ways of doing it, among them the Wallman, Stone-Čech and Alexandroff compactifications. Today we consider onl ...
... non-compact spaces, it is natural to ask the following question: can a given space be embedded homeomorphically in a compact space? The answer to this is always yes. In fact, there are many ways of doing it, among them the Wallman, Stone-Čech and Alexandroff compactifications. Today we consider onl ...
Compactness and total boundedness via nets The aim of this
... equipped with the coordinate-wise pre-ordering. It is easy to see that J is a directed set (verify this!). The function ψ : J → I, defined by ψ(α, B) = α, is “nondecreasing” and onto, an hence “tends to infinity”. Consequently, (xψ(α,B) )(α,B)∈J is a subnet of (xα ). Moreover, given A ∈ B, fix α0 ∈ ...
... equipped with the coordinate-wise pre-ordering. It is easy to see that J is a directed set (verify this!). The function ψ : J → I, defined by ψ(α, B) = α, is “nondecreasing” and onto, an hence “tends to infinity”. Consequently, (xψ(α,B) )(α,B)∈J is a subnet of (xα ). Moreover, given A ∈ B, fix α0 ∈ ...
PDF
... † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. ...
... † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. ...
Remarks concerning the invariance of Baire spaces under mappings
... Chapter 9, and [2]. Letfbe a mapping of a space P onto a space Q. Under what conditions onfmay we assert that if P is a Baire space then Q is a Baire space? It may be noticed that the image of a Baire space under a continuous mapping may fail to be a Baire space. Moreover, the image of a complete no ...
... Chapter 9, and [2]. Letfbe a mapping of a space P onto a space Q. Under what conditions onfmay we assert that if P is a Baire space then Q is a Baire space? It may be noticed that the image of a Baire space under a continuous mapping may fail to be a Baire space. Moreover, the image of a complete no ...
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.