STRUCTURED SINGULAR MANIFOLDS AND FACTORIZATION
... In particular, one can define topological spaces Emb(M, N ) and Sub(M, N ) of embeddings and submersions whose homotopy types reflect the smooth topology of M and N in a significant way; this is in the contrast with the space of all smooth maps Map(M, N ), the homotopy type of which is a homotopy in ...
... In particular, one can define topological spaces Emb(M, N ) and Sub(M, N ) of embeddings and submersions whose homotopy types reflect the smooth topology of M and N in a significant way; this is in the contrast with the space of all smooth maps Map(M, N ), the homotopy type of which is a homotopy in ...
The sequence selection properties of Cp(X)
... coordinate and let Un = {pn−1 (Um )}m∈ω . Then each Un is a−1 (n ∈ ω), take xn ∈ X − Ukn (n ∈ ω), then (xn ) ∈ / n∈ω pn (Ukn ). Hence {pn−1 (Ukn )}n∈ω is not a cover of X ω . Since ω X is homeomorphic to X, we can obtain clopen γ -covers of X as desired. 2 A space X is said to have the Hurewicz pro ...
... coordinate and let Un = {pn−1 (Um )}m∈ω . Then each Un is a−1 (n ∈ ω), take xn ∈ X − Ukn (n ∈ ω), then (xn ) ∈ / n∈ω pn (Ukn ). Hence {pn−1 (Ukn )}n∈ω is not a cover of X ω . Since ω X is homeomorphic to X, we can obtain clopen γ -covers of X as desired. 2 A space X is said to have the Hurewicz pro ...
On countable dense and strong n
... Hence a space X on which some topological group acts transitively is homogeneous. Is there for every homogeneous space X a nice topological group acting transitively on it? Nice because H(X) with the discrete topology acts transitively if X is homogeneous. ...
... Hence a space X on which some topological group acts transitively is homogeneous. Is there for every homogeneous space X a nice topological group acting transitively on it? Nice because H(X) with the discrete topology acts transitively if X is homogeneous. ...
On Totally sg-Continuity, Strongly sg
... reverse implication is not always true as shown in the following example. Example 3.2 Let X = {a, b, c } = Y , τ = { φ, {a}, {b}, {a, b}, X} and σ = {φ, {a}, Y}. Then the identity function f : (X, τ) → (Y, σ) is totally sg-continuous, but not strongly sg-continuous. Theorem 3.1 Every totally sg-cont ...
... reverse implication is not always true as shown in the following example. Example 3.2 Let X = {a, b, c } = Y , τ = { φ, {a}, {b}, {a, b}, X} and σ = {φ, {a}, Y}. Then the identity function f : (X, τ) → (Y, σ) is totally sg-continuous, but not strongly sg-continuous. Theorem 3.1 Every totally sg-cont ...
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.