TOPOLOGICAL SPACES
More specifically, a topological space is a set of points, along with a set of neighborhoods for each point, satisfying a set of axioms relating points and neighborhoods. A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Other spaces, such as Euclidean spaces, metric spaces and manifolds, are topological spaces with extra structures, properties or constraints.
Definition– A set X for which a topology T has been specified is called a topological space.
A topological space is an ordered pair (X, T) consisting of a set X and a topology T on X.
If X is a topological space with topology T, we say that a subset U of X is an open set of X if U belongs to collection of T.
Example- Let (X, T) be a topological space and Y a subset of X. Then,
S = { H ⊂ Y | H = G ∩ Y for some G ∈ T}
is a topology on Y. The open sets in Y are the intersections of the open sets in X with Y. This topology is called the induced or relative topology of Y in X, and (Y, S) is called a topological subspace of (X, T). For instance, the interval [0,1/2) is an open open subset of [0, 1] with respect to the induced metric topology of [0, 1] in R, since [0,1/2) = (-1/2, 1/2) ∩ [0, 1].
A set V ⊂ X is a neighborhood of a point x ∈ X if there exists an open set G ⊂ V with x ∈ G. We do not require that V itself is open. A topology T on X is called Hausdorff if every pair of distinct points x, y ∈ X has a pair of non-intersecting neighborhoods, meaning that there are neighborhoods Vx of x and Vy of y such that Vx ∩ Vx = ϕ. When the topology is clear, we often refer to X as a Hausdorff space. Almost all the topological spaces encountered in analysis are Hausdorff. For example, all metric topologies are Hausdorff.
Definition- A sequence (xn) in X converges to a limit x ∈ X if for every neighborhood V of x, there is a number N such that xn∈ V for all n ≥ N.
This definition says that the sequence eventually lies entirely in every neighborhood of x.
Definition- A function f : X → Y is continuous at x ∈ X if for each neighborhood W of f(x) there exists a neighborhood V of x such that f(V) ⊂ W. We say that f is continuous on X if it is continuous at every x ∈ X.
Theorem- Let (X, T) and (Y, S) be two topological spaces and f : X → Y.
Then f is continuous on X if and only if f-1 (G) ∈ T for every G ∈ S.
Definition- A function f : X → Y between topological spaces X and Y is a homeomorphism if it is a one to one, onto map and both f and f-1 are continuous. Two topological spaces X and Y are homeomorphic if there is a homeomorphism f : X → Y.
Homeomorphic spaces are indistinguishable as topological spaces. For example, if f : X → Y is a homeomorphism, then G is open in X if f(G) is open in Y, and a sequence (xn) converges to x in X if and only if the sequence (f(xn)) converges to f(x) in Y.
A one to one, onto map f always has an inverse f-1, but f-1 need not be continuous even if f is.
Example- We define f : [0, 2π) → T by f(Θ) = eiΘ, where [0, 2π) ⊂ R with the topology induced by the usual topology on R, and T ⊂ C is the unit circle with the topology induced by the usual topology on C. Then, ass illustrated in figure below, f is continuous but f-1 is not.
