Part II - Cornell Math
... (ii). Clearly, V := i U j still satifies condition (i), and W := m n+1 U j still satifies condition (ii) since the union of finite collection of compact sets is compact. Since X 0 − W is compact in a Hausdorff space X, it is closed, and so W = X \ (X 0 − W) is open in X. Thus, we have V ∩ W ⊂ T . (b ...
... (ii). Clearly, V := i U j still satifies condition (i), and W := m n+1 U j still satifies condition (ii) since the union of finite collection of compact sets is compact. Since X 0 − W is compact in a Hausdorff space X, it is closed, and so W = X \ (X 0 − W) is open in X. Thus, we have V ∩ W ⊂ T . (b ...
§2.1. Topological Spaces Let X be a set. A family T of subsets of X is
... (b) Let X be a set. Let T1 be the family of all subsets of X . Then T1 is a topology for X . It is called the discrete topology. It is the largest topology for X in the sense that if T is a topology for X , then T ⊂ T1 . (c) Let X be a metric space. Then the family of open subsets (defined in terms ...
... (b) Let X be a set. Let T1 be the family of all subsets of X . Then T1 is a topology for X . It is called the discrete topology. It is the largest topology for X in the sense that if T is a topology for X , then T ⊂ T1 . (c) Let X be a metric space. Then the family of open subsets (defined in terms ...
Continuity in topological spaces and topological invariance
... continuous on X if and only if ∀U ∈ υ, f −1 (U ) ∈ τ . Theorem 4. f is everywhere continuous on X if and only if f is continuous at every point x ∈ X. Proof. Assume that f is continuous at every point in X and let U ∈ υ. If f −1 (U ) = ∅, it is open. Otherwise, we may pick x ∈ f −1 (U ), and so f (x ...
... continuous on X if and only if ∀U ∈ υ, f −1 (U ) ∈ τ . Theorem 4. f is everywhere continuous on X if and only if f is continuous at every point x ∈ X. Proof. Assume that f is continuous at every point in X and let U ∈ υ. If f −1 (U ) = ∅, it is open. Otherwise, we may pick x ∈ f −1 (U ), and so f (x ...
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.