equivalent formulations for continuity
Suppose $f:X\to Y$ 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.

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  20130322 15:18:23 
Last modified on  20130322 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 