Disjoint union topology
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Revision as of 12:39, 9 May 2016 by Alec (Talk | contribs) (Created page with "{{Stub page|grade=A|msg=Grade A until it is more presentable.}} ==Definition== Suppose {{M|\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} }} be an indexed family of t...")
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Grade A until it is more presentable.
Definition
Suppose ((Xα,Jα))α∈I be an indexed family of topological spaces that are non-empty[1], the disjoint union topology is a topological space:
- with underlying set ∐α∈IXα, this is the disjoint union of sets, recall (x,β)∈∐α∈IXα⟺β∈I∧x∈Xβ and
- The topology where U∈P(∐α∈IXα) is considered open if and only if ∀α∈I[Xα∩U∈Jα][Note 1] - be sure to notice the abuse of notation going on here.
TODO: Flesh out notes, mention subspace Xα×{α} and such
Notes
- Jump up ↑ There's a very nasty abuse of notation going on here. First, note a set U is going to be a bunch of points of the form (x,γ) for various xs and γs (∈I). There is no "canonical projection" FROM the product to the spaces, as this would not be a function!
References
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TODO: Investigate the need to be non-empty, I suspect it's because the union "collapses" in this case, and the space wouldn't be a part of union
