# TOPOLOGICAL SPACES

A Topological Space, is,, A Geometrical Space in which Closeness is defined, but,, cannot necessarily be measured by a Numeric Distance.

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 V*

**∈**_{x }of x and V

_{y }of y such that V

_{x }

*∩*V

_{x }= 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 (x_{n}) in X converges to a limit x * ∈* X if for every neighborhood V of x, there is a number N such that x

_{n}

*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 (x_{n}) converges to x in X if and only if the sequence (f(x_{n})) 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^{iΘ}, 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.