# Continuity in Topological Spaces

Let be a function defined from topological space to topological space , then is said to be continuous at a point if for every neighborhood of , there exists a neighborhood of , such that .

In other words, let be two topological spaces. A function is said to be continuous at a point if and only if for every open set , which contains , there exists an open set , such that ; is the inverse image of .

It can also be defined as: let and be topological spaces. A function is said to be a continuous function at a point of if for any neighborhood of in , there is a neighborhood of in such that . The function is said to be a continuous function on if it is continuous at each point of .

**Note:** It may be noted that a function from topological space to topological space is said to be continuous on if is continuous of each point of .

**Theorems**

• A function from one topological space into another topological space is continuous if and only if for every open set in , is open in .

• A function from one topological space into another topological space is continuous if and only if for every closed set in , is closed in .

• If and are topological spaces, then a function is continuous on if and only if for any sub set of , .

• If and are topological spaces, then a function is continuous on if and only if for any sub set of , .

• If is an arbitrary topological space and is an indiscrete topological space, then every function is a continuous function on .

• Let and be topological spaces. If and are continuous mappings, then is continuous.