Definition talk:Topology/Definition 2

$\tau=\varnothing$ 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)


 * Okay. But the (relatively obscure) convention that $\bigcap\varnothing = S$ should be linked to then. Either way, it relies on informal notions like "universum" that have no solid basis, so I suggest not to use it. --barto (talk) 11:18, 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)