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 limitscontinuity, and connectedness. Other spaces, such as Euclidean spacesmetric 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 Vof x and Vy of y such that Vx V= 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) = e, 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. 



The interval and the circle are not homeomorphic.