testing continuity via filters

Proposition 1.

Let X,Y be topological spacesMathworldPlanetmath. Then a function f:XY is continuousMathworldPlanetmathPlanetmath iff it sends converging filters to converging filters.


Suppose first f is continuous. Let 𝔽 be a filter in X converging to x. We want to show that f(𝔽):={f(F)F𝔽} convergesPlanetmathPlanetmath to f(x). Let N be a neighborhoodMathworldPlanetmathPlanetmath of f(x). So there is an open set U such that f(x)UN. So f-1(U) is open and contains x, which means that f-1(U)𝔽 by assumptionPlanetmathPlanetmath. This means that ff-1(U)f(𝔽). Since ff-1(U)UN, we see that Nf(𝔽) as well.

Conversely, suppose f preserves converging filters. Let V be an open set in Y containing f(x). We want to find an open set U in X containing x, such that f(U)V. Let 𝔽 be the neighborhood filter of x. So 𝔽x. By assumption, f(𝔽)f(x). Since V is an open neighborhood of f(x), we have Vf(𝔽), or f(F)V for some F𝔽. Since F is a neighborhood of x, it contains an open neighborhood U of x. Furthermore, f(U)f(F)V. Since x is arbitrary, f is continuous. ∎

Title testing continuity via filters
Canonical name TestingContinuityViaFilters
Date of creation 2013-03-22 19:09:31
Last modified on 2013-03-22 19:09:31
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 4
Author CWoo (3771)
Entry type Result
Classification msc 26A15
Classification msc 54C05
Related topic Filter