Definition talk:Closure (Topology)/Definition 4

Circular Definition
This definition of the topological closure is dependent on the definition of the topological boundary which in turn is dependent on the definition of the topological closure. This doesn't seem right.

If this is not right, then the definition has to be removed and the part of the theorem Equivalence of Definitions of Closure of Topological Subspace that show the equivalence needs to be extracted as a separate theorem. --Leigh.Samphier (talk) 07:37, 11 May 2020 (EDT)


 * Is there a possible definition of the topological boundary which is itself not dependent upon the definition of topological closure?


 * I just found one in Fairchild and Ionescu-Tulcea, although they call it the Frontier of a set, because Boundary for them had another meaning in the theory of manifolds. I'll add this as an alternative definition for boundary, and note that some sources call it the Frontier. --Leigh.Samphier (talk) 04:59, 12 May 2020 (EDT)


 * Perhaps we need to refactor this. The original page defining topological closure bundled up a load of equivalences in the same page (it dated back from before we had imposed any sort of coherence on the site structure). Then a later evolution of the page extracted all those equivalences out as separate definitions (yeah I know, that was probably me). And subsequent to that, we embarked upon a policy where a result demonstrating an equivalence to a definition would not constitute a new definition page itself unless that definition could be sourced in some source work somewhere as an actual instance of how the entity can be defined.


 * But then this is not a circular definition if you adopt the idea that the definition of the topological boundary is dependent upon one of the other definitions of topological closure but not this one. --prime mover (talk) 09:58, 11 May 2020 (EDT)


 * I feel happier about it not being circular if there is another definition that is not dependent on the closure. So no need to refactor. --Leigh.Samphier (talk) 04:59, 12 May 2020 (EDT)