continuity and convergent nets


Let X and Y be topological spacesMathworldPlanetmath. A function f:XY is continuous at a point xX if and only if for each net (xα)αA in X converging to x, the net (f(xα))αA convergesPlanetmathPlanetmath to f(x).


If f is continuousMathworldPlanetmath, (xα)αA converges to x, and V is an open neighborhood of f(x) in Y, then f-1(V) is an open neighborhood of x in X, so there exists α0A such that xαf-1(V) for αα0. It follows that f(xα)V for αα0, hence that f(xα)f(x). Conversely, suppose there exists a net (xα)αA in X converging to x such that (f(xα))αA does not converge to f(x), so that, for some open subset V of Y containing f(x) and every α0A, there exists αα0A such that f(xα)V, hence such that xαf-1(V); as xαx by hypothesisMathworldPlanetmath, this implies that f-1(V) cannot be a neighborhoodMathworldPlanetmathPlanetmath of x, and thus that f fails to be continuous at x. ∎

Title continuity and convergent nets
Canonical name ContinuityAndConvergentNets
Date of creation 2013-03-22 18:37:53
Last modified on 2013-03-22 18:37:53
Owner azdbacks4234 (14155)
Last modified by azdbacks4234 (14155)
Numerical id 5
Author azdbacks4234 (14155)
Entry type Theorem
Classification msc 54A20
Related topic Net
Related topic Continuous