The $I$-adic topology on a commutative ring $A$ (with unity), where $I$ is an ideal of $A$. The closed sets are intersections of finite unions of sets of the form $a+I^n$ with $a\in A$ and $n\in\mathbb{N}$ (where $\mathbb{N}$ includes $0$). This topology has many trivial but very useful properties such as: The ring $A$ is separated (=Hausdorff) with respect to this topology if and only if $\displaystyle\bigcap_{n\in\mathbb{N}}I^n=0$. The most important example is the polynomial ring $A=B\left[X_1,X_2,...,X_n\right]$ with the ideal $I=\left(X_1,X_2,...,X_n\right)$. This one is separated, but not complete. Its completion is the ring of power series $B\left[\left[X_1,X_2,...,X_n\right]\right]$.

This is probably the most elementary example of a topology in algebra. I think Szamuely's book has more advanced ones.

The $I$-adic topology on a commutative ring $A$ (with unity), where $I$ is an ideal of $A$. The closed sets are intersections of finite unions of sets of the form $a+I^n$ with $a\in A$ and $n\in\mathbb{N}$ (where $\mathbb{N}$ includes $0$). This topology has many trivial but very useful properties such as: The ring $A$ is separated (=Hausdorff) with respect to this topology if and only if $\displaystyle\bigcap_{n\in\mathbb{N}}I^n=0$. The most important example is the polynomial ring $A=B\left[X_1,X_2,...,X_n\right]$ with the ideal $I=\left(X_1,X_2,...,X_n\right)$. This one is separated, but not complete. Its completion is the ring of power series $B\left[\left[X_1,X_2,...,X_n\right]\right]$. See Szamuely's notes on local algebra for more about this topology.

This is probably the most elementary example of a topology in algebra. I think Szamuely's book has more advanced ones.