**Definition**: Let (*X, T*) and *(Y, T ^{*}) *be topological spaces. A function f from

*X*into

*Y*is continuous relative to

*T*and

*T*Or,

^{*},*T-T*continuous or simply continuous, if and only if the inverse image

^{*}*f*of every

^{-1}[H]*T*open subset

^{*}*H*of

*Y*is a

*T*– open subset of

*X*, that is, if and only if

* H **∊**T ^{*} *implies

*f*[

^{-1}*H*]

*∊*

*T.*

**Example**: Consider the following topologies on

X = {a, b, c, d} and Y = {x, y, z, w} respectively:

T = {X, ∅, {a}, {a, b}, {a, b, c}}, T^{*} = {Y, ∅, {x}, {y}, {x, y}, {y, z,w}}

Also consider the function *f:X**→**Y *and *g: X **→** Y *defined by the diagrams

The function *f *is continuous since the inverse of each member of the topology *T ^{*} *on

*Y*is a member of the topology

*T*on

*X*. The function g is not continuous since

*{y, z, w} **∊** T ^{*}, *i.e., is an open subset of

*Y*, but its inverse image

* g ^{-1}[{y, z, w}] = {c, d} *is not an open subset of

*X*, i.e., does not belong to

*T*.

**Theorem: Let the function f: X → Y and g: X→Y be continuous. Then the composition function g ^{o} f : X →Z is also continuous.**

**Proof**: Let *G* be an open subset of *Z*. Then *g ^{-1}*[

*G*] is open in

*Y*since g is continuous. But

*f*is also continuous, so

*f*[

^{-1}*g*[

^{-1}*G*]] is open in X. Now

*g*]

^{o}f*[*

^{-1}*G*]

*= f*[

^{-1}*g*[

^{-1}*G*]]

Thus *f ^{-1}*[

*g*[

^{-1}*G*]] is open in

*X*for every open subset

*G*of

*Z*, or,

*g*is continuous. (Proved)

^{o}f**Theorem: A function f: X→Y is continuous if and only if the inverse image of every closed subset of Y is a closed subset of X.**

**Proof**: Suppose *f: X**→**Y *is continuous, and let *F* be a closed subset of *Y*. Then *F ^{c}* is open, and so

*f*[

^{ -1}*F*] is open in

^{c}*X*. But

*f*[

^{ -1}*F*] = (

^{c}*f*[

^{ -1}*F*])

^{c}; therefore

*f*[

^{ -1}*F*] is closed.

Conversely, assume *F* closed in *Y *implies *f ^{ -1}*[

*F*] closed in

*X*. Let G be an open subset of

*Y*. Then

*G*is closed in

^{c}*Y,*and so

*f*[

^{ -1}*G*] = (

^{c}*f*[

^{ -1}*F*])

^{c}is closed in

*X*. Accordingly,

*f*[

^{ -1}*F*] is open and therefore

*f*is continuous. (Proved)

**Theorem: Let X and Y be topological spaces. Then a function f:X→Y is continuous if and only if it is continuous at every point p ∊ X.**

Proof: Assume f is continuous, and let *H* ⊂* Y *be an open set containing *f(p). *But then *p **∊** f ^{–1 }*[

*H*], and

*f*[

^{–1 }*H*] is open. Hence

*f*is continuous at

*p*.

Now suppose f is continuous at every point *p **∊** X, *and let *H **⊂** Y *be open. For every *p **∊** f ^{ – 1 }*[

*H*], there exists an open set

*G*⊂

_{p}*X*such that

*p*

*∊*

*G*⊂

_{p}*f*[

^{–1 }*H*]. Hence

*f*

^{–}^{1 }[

*H*] = ⋃ {

*G*

_{p}: p∊*f*[

^{–1 }*H*]} a union of open sets. Accordingly,

*f*[

^{–1 }*H*] is open and so f is continuous. (Proved)