equivalent formulations for continuity

Suppose f:XY is a function between topological spacesMathworldPlanetmath X, Y. Then the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    f is continuousMathworldPlanetmathPlanetmath.

  2. 2.

    If B is open in Y, then f-1(B) is open in X.

  3. 3.

    If B is closed in Y, then f-1(B) is closed in X.

  4. 4.

    f(A¯)f(A)¯ for all AX.

  5. 5.

    If (xi) is a net in X converging to x, then (f(xi)) is a net in Y converging to f(x). The concept of net can be replaced by the more familiar one of sequence if the spaces X and Y are first countable.

  6. 6.

    Whenever two nets S and T in X convergePlanetmathPlanetmath to the same point, then fS and fT converge to the same point in Y.

  7. 7.

    If 𝔽 is a filter on X that converges to x, then f(𝔽) is a filter on Y that converges to f(x). Here, f(𝔽) is the filter generated by the filter base {f(F)F𝔽}.

  8. 8.

    If B is any element of a subbase (http://planetmath.org/Subbasis) 𝒮 for the topology of Y, then f-1(B) is open in X.

  9. 9.

    If B is any element of a basis for the topology of Y, then f-1(B) is open in X.

  10. 10.

    If xX, and N is any neighborhood of f(x), then f-1(N) is a neighborhood of x.

  11. 11.

    f is continuous at every point in X.

Title equivalent formulations for continuity
Canonical name EquivalentFormulationsForContinuity
Date of creation 2013-03-22 15:18:23
Last modified on 2013-03-22 15:18:23
Owner matte (1858)
Last modified by matte (1858)
Numerical id 14
Author matte (1858)
Entry type Theorem
Classification msc 26A15
Classification msc 54C05
Related topic Characterization