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 functionf: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 Xto 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 functionf: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.

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