Open set

2020 Mathematics Subject Classification: Primary: 54A05 [MSN][ZBL]

in a topological space

An element of the topology (cf. Topological structure (topology)) of this space. More specifically, let the topology $\tau$ of a topological space $(X, \tau)$ be defined as a system $\tau$ of subsets of the set $X$ such that:

1. $X\in\tau$, $\emptyset\in\tau$;
2. if $O_i\in\tau$, where $i=1,2$, then $O_1\cap O_2\in\tau$;
3. if $O_{\alpha}\in\tau$, where $\alpha\in\mathfrak{A}$, then $\bigcup\{O_{\alpha} : \alpha\in\mathfrak{A} \} \in \tau$.

The open sets in the space $(X, \tau)$ are then the elements of the topology $\tau$ and only them.

References

[a1]	R. Engelking, "General topology" , Heldermann (1989)