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

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

9. If $B$ is any element of a basis $ℬ$ 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 $X$.