Compact covering mappings and cofinal families of compact subsets
... mapping from a Π11 space onto a Π11 space is inductively perfect. (d) In Gödel’s universe L, there exists a compact covering mapping f : X → Y between two Borel spaces which is not inductively perfect; moreover X can be chosen to be the intersection of a Π02 and a Σ02 set, and Y can be chosen to be ...
... mapping from a Π11 space onto a Π11 space is inductively perfect. (d) In Gödel’s universe L, there exists a compact covering mapping f : X → Y between two Borel spaces which is not inductively perfect; moreover X can be chosen to be the intersection of a Π02 and a Σ02 set, and Y can be chosen to be ...
Fourier analysis on abelian groups
... We now come to the first really non-trivial result about these spaces. There are two equivalent forms of this result. The first is phrased in terms of maximal translationinvariant subspaces: Proposition 1.3 (Gelfand-Mazur theorem, special case). All maximal translationinvariant subspaces are hyperpl ...
... We now come to the first really non-trivial result about these spaces. There are two equivalent forms of this result. The first is phrased in terms of maximal translationinvariant subspaces: Proposition 1.3 (Gelfand-Mazur theorem, special case). All maximal translationinvariant subspaces are hyperpl ...
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... A function f : (X, τ) → (Y, σ) is said to be semicontinuous[9] (resp. α-continuous [12], pre-continuous [11], totally continuous [7], totally semi-continuous [16]) if the inverse image of every open subset of (Y, σ) is a semi-open (resp. α-open, preopen, clopen, semi-clopen) subset of (X,τ). Definit ...
... A function f : (X, τ) → (Y, σ) is said to be semicontinuous[9] (resp. α-continuous [12], pre-continuous [11], totally continuous [7], totally semi-continuous [16]) if the inverse image of every open subset of (Y, σ) is a semi-open (resp. α-open, preopen, clopen, semi-clopen) subset of (X,τ). Definit ...
Topological Vector Spaces and Continuous Linear Functionals
... be a (real or complex) vector space, and let {ρν } be a collection of seminorms on X that separates the nonzero points of X from 0 in the sense that for each x 6= 0 there exists a ν such that ρν (x) > 0. For each y ∈ X and each index ν, define gy,ν (x) = ρν (x − y). Then X, equipped with the weakest ...
... be a (real or complex) vector space, and let {ρν } be a collection of seminorms on X that separates the nonzero points of X from 0 in the sense that for each x 6= 0 there exists a ν such that ρν (x) > 0. For each y ∈ X and each index ν, define gy,ν (x) = ρν (x − y). Then X, equipped with the weakest ...
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.