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equivalent formulations for continuity∗
2013-03-21 19:16:36
Suppose f : X → Y is a function between topological spaces X, Y . Then
the following are equivalent:
1. f is continuous.
2. If B is open in Y , then f −1 (B) is open in X.
3. If B is closed in Y , then f −1 (B) is closed in X.
4. f A ⊆ f (A) for all A ⊆ X.
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. Whenever two nets S and T in X converge to the same point, then f ◦ S
and f ◦ T converge to the same point in Y .
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 ) | F ∈ F}.
8. If B is any element of a subbase S for the topology of Y , then f −1 (B) is
open in X.
9. If B is any element of a basis B for the topology of Y , then f −1 (B) is
open in X.
10. If x ∈ X, and N is any neighborhood of f (x), then f −1 (N ) is a neighborhood of x.
11. f is continuous at every point in X.
∗ hEquivalentFormulationsForContinuityi
created: h2013-03-21i by: hmattei version:
h37106i Privacy setting: h1i hTheoremi h26A15i h54C05i
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