Post History
#3: Post edited
by
clemens
·
2026-03-07T05:40:34Z (6 months ago)
Why is the union of topologies not in general a topology?
- Why is the union of two topologies not in general a topology?
#2: Post edited
by
clemens
·
2026-03-07T05:40:25Z (6 months ago)
Why is the union of topologies not a topology?
- Why is the union of topologies not in general a topology?
#1: Initial revision
by
clemens
·
2026-03-07T05:37:18Z (6 months ago)
Why is the union of topologies not a topology?
(Note: this question is based on [this question](https://math.stackexchange.com/questions/2713668/a-union-of-topologies-is-a-topology/5124173#5124173) from the other website, written by user390960. The answer was previously written by me.)
I read that the intersection of two topologies $\mathcal{T}_1$ and $\mathcal{T}_2$ is always a topology, which makes sense; but I also read that the union of two topologies is not in general a topology, because the intersection of an open in $\mathcal{T}_1$ and an open in $\mathcal{T}_2$ is not necessarily open.
How can this be? Shouldn't the two open sets in $\mathcal{T}_1$ and $\mathcal{T}_2$ be disjoint and hence have an open (empty) intersection?