# Continuity in Topological Spaces

Let $$f$$ be a function defined from topological space $$X$$ to topological space $$Y$$, then $$f$$ is said to be continuous at a point $$x \in X$$ if for every neighborhood $$V$$ of $$f\left( x \right)$$, there exists a neighborhood $$U$$ 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 exists 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 neighborhood $${N_Y}$$ of $$f\left( {{x_o}} \right)$$ in $$Y$$, there is a neighborhood $${N_X}$$ 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 point 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.