Irreducibility of product spaces with finitely many points removed
... We prove an induction step that can be used to show that in certain cases the removal of finitely many points from a product space produces an irreducible space. For example, we show that whenever γ is less than ℵω , removing finitely many points from the product of γ many first countable compact sp ...
... We prove an induction step that can be used to show that in certain cases the removal of finitely many points from a product space produces an irreducible space. For example, we show that whenever γ is less than ℵω , removing finitely many points from the product of γ many first countable compact sp ...
16. Maps between manifolds Definition 16.1. Let f : X −→ Y be a
... Theorem 16.9. Let f : X −→ Y be a local homeomorphism of manifolds. Then f has the lifting property if and only if f is an unramified cover. We have already shown one direction of (16.9). To prove the other direction we need the following basic result: Theorem 16.10. Let f : X −→ Y be a local homeo ...
... Theorem 16.9. Let f : X −→ Y be a local homeomorphism of manifolds. Then f has the lifting property if and only if f is an unramified cover. We have already shown one direction of (16.9). To prove the other direction we need the following basic result: Theorem 16.10. Let f : X −→ Y be a local homeo ...
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
... If B is a subspace of X such that A B A , then show that B is connected. (b)(i) Show that a topological space X is disconnected there exists a continuous mapping of X onto the discrete twopoint space {0, 1}. (ii) Prove that the product of any nonempty class of connected spaces is ...
... If B is a subspace of X such that A B A , then show that B is connected. (b)(i) Show that a topological space X is disconnected there exists a continuous mapping of X onto the discrete twopoint space {0, 1}. (ii) Prove that the product of any nonempty class of connected spaces is ...
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.