Closed Subset of a Topological Space

Intuitively, a closed set of a Topological Space is a set that "contains its boundary". In Classical Mathematics, there are 4 equivalent ways to formalize this intuition:

1. A subset $S\subseteq X$ is closed if its Complement $X\backslash S$ is open.
2. A subset $S\subseteq X$ is closed if there is some open set $U\in\mathcal{O}({X})$ such that $S=X\backslash U$.
3. A subset $S\subseteq X$ is closed if it contains all of its Adherent Points.
4. A subset $S\subseteq X$ is closed if it contains all of its Accumulation Points.

In Constructive Mathematics, these three notions are different. Defininition 2 is the strongest, and is far too strong: even the closed intervals $[a,b]\subseteq\mathbb{R}$ do not satisfy this!

References

• https://ncatlab.org/nlab/show/closed+subspace#constructive_mathematics