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Why is the union of two topologies not in general a topology?
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(Note: this question is based on this question 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?
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