Remedial topology
... Definition 1.1. A set of all subsets of M is denoted 2M . Topology on M is a collection of subsets S ⊂ 2M called open subsets, and satisfying the following conditions. 1. Empty set and M are open 2. A union of any number of open sets is open 3. An intersection of a finite number of open subsets is o ...
... Definition 1.1. A set of all subsets of M is denoted 2M . Topology on M is a collection of subsets S ⊂ 2M called open subsets, and satisfying the following conditions. 1. Empty set and M are open 2. A union of any number of open sets is open 3. An intersection of a finite number of open subsets is o ...
TOPOLOGY 1. Introduction By now, we`ve seen many uses of
... Another topology that can be defined on any set X (finite or infinite) is the cofinite topology. In the cofininte topology, open sets are defined to be those subsets U ⊂ X such that the complement of U in X is finite (alternatively, the closed sets are the finite sets). Note that if X is finite, the ...
... Another topology that can be defined on any set X (finite or infinite) is the cofinite topology. In the cofininte topology, open sets are defined to be those subsets U ⊂ X such that the complement of U in X is finite (alternatively, the closed sets are the finite sets). Note that if X is finite, the ...
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.