# equivalent formulations for continuity

Suppose $f\colon X\to Y$ is a function between topological spaces $X$, $Y$. Then the following are equivalent:

1. 1.

$f$ is continuous.

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.

$f\!\left(\overline{A}\right)\subseteq\overline{f(A)}$ for all $A\subseteq X$.

5. 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. 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. 7.

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

8. 8.

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

9. 9.

If $B$ is any element of a basis $\mathcal{B}$ for the topology of $Y$, then $f^{-1}(B)$ is open in $X$.

10. 10.

If $x\in X$, 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 EquivalentFormulationsForContinuity 2013-03-22 15:18:23 2013-03-22 15:18:23 matte (1858) matte (1858) 14 matte (1858) Theorem msc 26A15 msc 54C05 Characterization