Difference between revisions of "Disjoint union topology"
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* The [[topology]] where {{M|U\in\mathcal{P}(\coprod_{\alpha\in I}X_\alpha)}} is considered [[open set|open]] ''if and only if'' {{M|1=\forall \alpha\in I[X_\alpha\cap U\in\mathcal{J}_\alpha]}}<ref group="Note">There's a very nasty abuse of notation going on here. First, note a set {{M|U}} is going to be a bunch of points of the form {{M|(x,\gamma)}} for various {{M|x}}s and {{M|\gamma}}s ({{M|\in I}}). There is no "canonical projection" FROM the product to the spaces, as this would not be a [[function]]!</ref> - be sure to notice the abuse of notation going on here. | * The [[topology]] where {{M|U\in\mathcal{P}(\coprod_{\alpha\in I}X_\alpha)}} is considered [[open set|open]] ''if and only if'' {{M|1=\forall \alpha\in I[X_\alpha\cap U\in\mathcal{J}_\alpha]}}<ref group="Note">There's a very nasty abuse of notation going on here. First, note a set {{M|U}} is going to be a bunch of points of the form {{M|(x,\gamma)}} for various {{M|x}}s and {{M|\gamma}}s ({{M|\in I}}). There is no "canonical projection" FROM the product to the spaces, as this would not be a [[function]]!</ref> - be sure to notice the abuse of notation going on here. | ||
{{Todo|Flesh out notes, mention subspace {{M|X_\alpha\times\{\alpha\} }} and such}} | {{Todo|Flesh out notes, mention subspace {{M|X_\alpha\times\{\alpha\} }} and such}} | ||
| + | '''Claim 1: ''' this is indeed a topology | ||
| + | {{Todo|Define the [[canonical injections of the disjoint union topology]] here}} | ||
| + | Note that [[the canonical injections of the disjoint union topology are topological embeddings]] | ||
| + | ==[[Characteristic property of the disjoint union topology|Characteristic property]]== | ||
| + | {{:Characteristic property of the disjoint union topology/Statement}} | ||
| + | ==Proof of claims== | ||
| + | {{Requires proof|msg=Actually surprisingly easy to prove, done on paper. page 1, 7/8/2016, Intro to top manifolds notes. Filed}} | ||
==Notes== | ==Notes== | ||
<references group="Note"/> | <references group="Note"/> | ||
Latest revision as of 23:47, 25 September 2016
Definition
Suppose [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath] be an indexed family of topological spaces that are non-empty[1], the disjoint union topology is a topological space:
- with underlying set [ilmath]\coprod_{\alpha\in I}X_\alpha[/ilmath], this is the disjoint union of sets, recall [ilmath](x,\beta)\in\coprod_{\alpha\in I}X_\alpha\iff \beta\in I\wedge x\in X_\beta[/ilmath] and
- The topology where [ilmath]U\in\mathcal{P}(\coprod_{\alpha\in I}X_\alpha)[/ilmath] is considered open if and only if [ilmath]\forall \alpha\in I[X_\alpha\cap U\in\mathcal{J}_\alpha][/ilmath][Note 1] - be sure to notice the abuse of notation going on here.
TODO: Flesh out notes, mention subspace [ilmath]X_\alpha\times\{\alpha\} [/ilmath] and such
Claim 1: this is indeed a topology
TODO: Define the canonical injections of the disjoint union topology here
Note that the canonical injections of the disjoint union topology are topological embeddings
Characteristic property
Let [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath] be a collection of topological spaces and let [ilmath](Y,\mathcal{ K })[/ilmath] be another topological space]]. We denote by [ilmath]\coprod_{\alpha\in I}X_\alpha[/ilmath] the disjoint unions of the underlying sets of the members of the family, and by [ilmath]\mathcal{J} [/ilmath] the disjoint union on it (so [ilmath](\coprod_{\alpha\in I}X_\alpha,\mathcal{J})[/ilmath] is the disjoint union topological construct of the [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath] family) and lastly, let [ilmath]f:\coprod_{\alpha\in I}X_\alpha\rightarrow Y[/ilmath] be a map (not necessarily continuous) then:
we state the characteristic property of disjoint union topology as follows:
TODO: rewrite and rephrase this
- [ilmath]f:\coprod_{\alpha\in I}X_\alpha\rightarrow Y[/ilmath] is continuous if and only if [ilmath]\forall\alpha\in I\big[f\big\vert_{X_\alpha^*}:i_\alpha({X_\alpha})\rightarrow Y\text{ is continuous}\big][/ilmath]
Where (for [ilmath]\beta\in I[/ilmath]) we have [ilmath]i_\beta:X_\beta\rightarrow\coprod_{\alpha\in I}X_\alpha[/ilmath] given by [ilmath]i_\beta:x\mapsto(\beta,x)[/ilmath] are the canonical injections
Proof of claims
The message provided is:
Notes
- ↑ There's a very nasty abuse of notation going on here. First, note a set [ilmath]U[/ilmath] is going to be a bunch of points of the form [ilmath](x,\gamma)[/ilmath] for various [ilmath]x[/ilmath]s and [ilmath]\gamma[/ilmath]s ([ilmath]\in I[/ilmath]). 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
