equivalent formulations for continuity

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 $\mathbb{F}$ 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.

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.

If $B$ is any element of a basis $\mathcal{B}$ for the topology of $Y$ , then $f^{-1}(B)$ is open in $X$ .
10.

If $x\in X$ , 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 $X$ .

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