Definition talk:Topology/Definition 2
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$\tau=\varnothing$ satisfies this definition. Or am I overlooking something? --barto (talk) 10:07, 24 October 2017 (EDT)
- Yes, this is correct. The set $\tau$ is the set of all open sets of $T$. The union of any number of open sets of $T$ is an open set of $T$. That is, the union of any arbitrary subset of $\tau$ is an element of $\tau$. Similar for intersections. It is NOT the case that the union of any arbitrary subsets of $S$ is an open set of $T$.
- $\tau = \varnothing$ does NOT satisfy the definition, as the intersection of any arbitrary subset of $\tau$ is also in $\tau$, and the intersection of $\varnothing$ sets equals $S$, so $S \in \tau$ for a start. --prime mover (talk) 10:59, 24 October 2017 (EDT)
- It is linked to, in the Equivalence proof. I would be reluctant to dismiss a definition, backed up in the literature, because it rests upon a convention that is considered by some to be "obscure" and "informal". --prime mover (talk) 11:34, 24 October 2017 (EDT)