Compactness - GMU Math 631 Spring 2011
... Theorem 23. Let X be a T1 space. Then the following conditions are equivalent: (1) X is countably compact; (2) Every infinite subset of X has a limit point in X;11 (3) X does not contain infinite closed discrete subspaces. Sketch of proof. Let D be an infinite closed discrete subspace of X. We can a ...
... Theorem 23. Let X be a T1 space. Then the following conditions are equivalent: (1) X is countably compact; (2) Every infinite subset of X has a limit point in X;11 (3) X does not contain infinite closed discrete subspaces. Sketch of proof. Let D be an infinite closed discrete subspace of X. We can a ...
Maximal Tychonoff Spaces and Normal Isolator Covers
... a topological space (X, τ ) is maximal P provided that τ is a maximal element in A(X). In [6], it had been shown that a topological space (X, τ ) is maximal P if and only if every continuous bijection from a space (Y, τ1 ) with the property P to (X, τ ) is a homeomorphism. In 1943 Hewitt [15] and in ...
... a topological space (X, τ ) is maximal P provided that τ is a maximal element in A(X). In [6], it had been shown that a topological space (X, τ ) is maximal P if and only if every continuous bijection from a space (Y, τ1 ) with the property P to (X, τ ) is a homeomorphism. In 1943 Hewitt [15] and in ...
Groupoids in categories with pretopology
... with equal range or equal source, respectively. We explain in Section 3 how such elementwise formulas should be interpreted in a general category, following ideas from synthetic geometry [22, 30]. We use elementwise formulas throughout because they clarify statements and proofs. We define functors b ...
... with equal range or equal source, respectively. We explain in Section 3 how such elementwise formulas should be interpreted in a general category, following ideas from synthetic geometry [22, 30]. We use elementwise formulas throughout because they clarify statements and proofs. We define functors b ...
Compact matrix operators on a new sequence space related to ℓp
... Let S and M be subsets of a metric space (X, d) and ε > . Then S is called an ε-net of M in X if for every x ∈ M there exists s ∈ S such that d(x, s) < ε. Further, if the set S is finite, then the ε-net S of M is called a finite ε-net of M, and we say that M has a finite ε-net in X. A subset M of a me ...
... Let S and M be subsets of a metric space (X, d) and ε > . Then S is called an ε-net of M in X if for every x ∈ M there exists s ∈ S such that d(x, s) < ε. Further, if the set S is finite, then the ε-net S of M is called a finite ε-net of M, and we say that M has a finite ε-net in X. A subset M of a me ...
Math 55a - Harvard Mathematics
... NB we diverge here from the order of presentation in Rudin, where continuity is postponed until Chapter 4. Continuity of functions between metric spaces. In a typical mathematical theory one is concerned not only with the objects and their properties, but also (and perhaps even more importantly) wit ...
... NB we diverge here from the order of presentation in Rudin, where continuity is postponed until Chapter 4. Continuity of functions between metric spaces. In a typical mathematical theory one is concerned not only with the objects and their properties, but also (and perhaps even more importantly) wit ...
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.