# Continuity in Topological Spaces

Let $f$ be a function define from topological space $X$ to topological space $Y$, then $f$ is said to be continuous at a point $x \in X$ if every neighbourhood $V$ of $f\left( x \right)$, there exist a neighbourhood $U$ of $x$, such that $f\left( U \right) \subseteq V$.

In other words, let $X,Y$ be two topological spaces. A function $f:X \to Y$ is said to be continuous at a point $x \in X$ if and only if for every open set $V$, which contains $f\left( x \right) \in Y$, there exist an open set $U$, such that $x \in U \subseteq {f^{ - 1}}\left( V \right)$; ${f^{ - 1}}\left( V \right)$ is the inverse image of $V$.

It can also be defined as, let $\left( {X,{\tau _X}} \right)$ and $\left( {X,{\tau _Y}} \right)$ be topological spaces. A function $f:X \to Y$ is said to be a continuous function at a point ${x_o}$ of $X$ if for any neighbourhood ${N_Y}$ of $f\left( {{x_o}} \right)$ in $Y$, there is a neighbourhood ${N_X}$ of ${x_o}$ in $X$ such that $f\left( {{N_X}} \right) \subseteq {N_Y}$. The function$f:X \to Y$ is said to be a continuous function on $X$ if it is continuous at each point of $X$.

Note: It may be noted that a function $f$ from topological space $X$to topological space $Y$ is said to be continuous on $X$ if $X$ is continuous of each points of $X$.

Theorems:
• A function $f$ from one topological space $X$ into another topological space $Y$ is continuous if and only if for every open set $V$ in $Y$, ${f^{ - 1}}\left( V \right)$ is open in $X$.
• A function $f$ from one topological space $X$ into another topological space $Y$ is continuous if and only if for every closed set $C$ in $Y$, ${f^{ - 1}}\left( C \right)$ is closed in $X$.
• If $X$ and $Y$ are topological spaces, then a function $f:X \to Y$ is continuous on $X$ if and only if for any sub set $A$ of $X$, $f\left( {\overline A } \right) \subseteq \overline {f\left( A \right)}$.
• If $X$ and $Y$ are topological spaces, then a function $f:X \to Y$ is continuous on $X$ if and only if for any sub set $A$ of $X$, ${f^{ - 1}}\left( {{A^o}} \right) \subseteq {\left[ {{f^{ - 1}}\left( A \right)} \right]^o}$.
• If $X$ is an arbitrary topological space and $Y$ is an indiscrete topological space, then every function$f:X \to Y$ is a continuous function on $X$.
• Let $X,Y$ and $Z$ be topological spaces. If $f:X \to Y$ and $g:Y \to Z$ are continuous mappings, then $g \circ f:X \to Z$ is continuous.