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. $f\!\left(\closure{A}\right)\subseteq\closure{f(A)}$ 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 $X$ .