: Definition ∈
... Proof. Let x ∈ C. Then U = X is a neighborhood of x, so by local connectivity there is a connected neighborhood V of x with V ⊂ U = X. Since V is connected, V ⊂ C. Hence C contains a neighborhood around each of its points, and must be open. Theorem: the image of a locally connected space which is bo ...
... Proof. Let x ∈ C. Then U = X is a neighborhood of x, so by local connectivity there is a connected neighborhood V of x with V ⊂ U = X. Since V is connected, V ⊂ C. Hence C contains a neighborhood around each of its points, and must be open. Theorem: the image of a locally connected space which is bo ...
Lecture 6 outline copy
... • The notion of a function being continuous (with respect to a given topology) brings the open sets into the story as the open sets distinguish a certain subset of functions. • This subset of functions are called continuous. • Suppose X, Y are topological spaces: A function ƒ: X → Y is said to be co ...
... • The notion of a function being continuous (with respect to a given topology) brings the open sets into the story as the open sets distinguish a certain subset of functions. • This subset of functions are called continuous. • Suppose X, Y are topological spaces: A function ƒ: X → Y is said to be co ...
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... From the definition, it is immediately clear that any discrete space is door. To find more examples, let us look at the singletons of a door space X. For each x ∈ X, either {x} is open or closed. Call a point x in X open or closed according to whether {x} is open or closed. Let A be the collection o ...
... From the definition, it is immediately clear that any discrete space is door. To find more examples, let us look at the singletons of a door space X. For each x ∈ X, either {x} is open or closed. Call a point x in X open or closed according to whether {x} is open or closed. Let A be the collection o ...
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... of A, if any, of A if A is endowed with (i) usual topology of reals, (ii) discrete topology, (iii) cofinite topology. ...
... of A, if any, of A if A is endowed with (i) usual topology of reals, (ii) discrete topology, (iii) cofinite topology. ...
Algebraic Topology Lecture 1
... Definition 2.2 (Homotopy Equivalence). Two topological spaces X, Y are homotopy equivalent if there exists continuous maps f : X → Y and g : Y → X such that f ◦ g ∼ Id and g ◦ f ∼ Id, that is homotopic to the identity map, we then write X 'Y. Remark 2.3. If f ◦ g and g ◦ f are not only homotopic to ...
... Definition 2.2 (Homotopy Equivalence). Two topological spaces X, Y are homotopy equivalent if there exists continuous maps f : X → Y and g : Y → X such that f ◦ g ∼ Id and g ◦ f ∼ Id, that is homotopic to the identity map, we then write X 'Y. Remark 2.3. If f ◦ g and g ◦ f are not only homotopic to ...
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... The long line (without initial point) is homogeneous, but it is not bihomogeneous as its self-homeomorphisms are all order-preserving. This can be considered a pathological example, as most homogeneous topological spaces encountered in practice are also bihomogeneous. Every topological group is biho ...
... The long line (without initial point) is homogeneous, but it is not bihomogeneous as its self-homeomorphisms are all order-preserving. This can be considered a pathological example, as most homogeneous topological spaces encountered in practice are also bihomogeneous. Every topological group is biho ...
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.