Tutorial Continuity in Topological Spaces


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 belongs to X if every neighbourhood V of f(x), there exist a neighbourhood V of x, such that .

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

It can also be defined as, let (X,Tx) and (Y,Ty) be topological spaces. A function is said to be a continuous function at a point xo of X if for any neighbourhood N of f(xo) in Y, there is a neighbourhood N of xo in X such that . The function 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 ffrom 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 Vin Y, 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 in , is closed in .
    • If X and Y are topological spaces, then a function is continuous on X if and only if for any sub set A of X, .
    • If X and Y are topological spaces, then a function is continuous on X if and only if for any sub set A of X, .
    • If X is an arbitrary topological space and is an indiscrete topological space, then every function is a continuous function on X.
    • Let X,Y and Z be topological spaces. If and are continuous mappings, then is continuous.



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