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[parent] equivalent formulations for continuity (Theorem)

Suppose $ f\colon X\to 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\!\left(\overline {A}\right)\subseteq\overline {f(A)}$ for all $ A\subseteq X$.
  5. If $ (x_i)$ is a net in $ X$ converging to $ x$, then $ (f(x_i))$ 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 \circ S$ and $ f \circ T$ converge to the same point in $ Y$.
  7. If $ B$ is any element of a subbase $ \mathbb{B}$ for the topology of $ Y$, then $ f^{-1}(B)$ is open in $ X$.
  8. If $ x \in X$, and $ N$ is any neighborhood of $ f(x)$, then $ f^{-1}(N)$ is a neighborhood of $ x$.
  9. $ f$ is continuous at every point in $ X$.



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Cross-references: continuous at, neighborhood, point, converge, first countable, sequence, net, closed, open, continuous, the following are equivalent, topological spaces, function
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This is version 8 of equivalent formulations for continuity, born on 2005-05-23, modified 2007-11-24.
Object id is 7106, canonical name is EquivalentFormulationsForContinuity.
Accessed 1487 times total.

Classification:
AMS MSC: 54C05 (General topology :: Maps and general types of spaces defined by maps :: Continuous maps)
 26A15 (Real functions :: Functions of one variable :: Continuity and related questions )
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