The canonical injections of the disjoint union topology are topological embeddings

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Important for progress!

Statement

Let [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath] be a collection of topological spaces and let [ilmath](\coprod_{\alpha\in I}X_\alpha,\mathcal{J})[/ilmath] be the disjoint union space of that family. With this construction we get some canonical injections:

• For each [ilmath]\beta\in I[/ilmath] we get a map (called a canonical injection) [ilmath]i_\beta:X_\beta\rightarrow\coprod_{\alpha\in I}X_\alpha[/ilmath] given by [ilmath]i_\beta:x\mapsto(\beta,x)[/ilmath]

We claim that each [ilmath]i_\beta[/ilmath] is a topological embedding (that means [ilmath]i_\beta[/ilmath] is injective and continuous and a homeomorphism between [ilmath]X_\beta[/ilmath] and [ilmath]i_\beta(X_\beta)[/ilmath] (its image))

Proof

Let [ilmath]\beta\in I[/ilmath] be given.

• The proof that [ilmath]i_\beta:X_\beta\rightarrow\coprod_{\alpha\in I}X_\alpha[/ilmath] by [ilmath]i_\beta:x\mapsto(\beta,x)[/ilmath] consists of three parts:
  1. Continuity of [ilmath]i_\beta[/ilmath] - covered on the canonical injections of the disjoint union topology page so not shown on this pag
  2. [ilmath]i_\beta[/ilmath] being injective and
  3. [ilmath]i_\beta[/ilmath] being a homeomorphism between [ilmath]X_\beta[/ilmath] and [ilmath]i_\beta(X_\beta)[/ilmath]

Grade: A

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Done on paper, it isn't hard. I want to save my work now though