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equivalent formulations for continuity∗ matte† 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 † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1