Canonical injection of the subspace topology

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I have documented the canonical injection on the subspace topology page and mentioned its use there, this page however is supposed to contain more information, like two proofs of continuity (one directly, one as a corollary to the characteristic property of the subspace topology

Definition

The following is temporary:

• See the definition part of the subspace topology page - bottom 2 paragraphs cover it

We claim here it is continuous and elaborate on the note left on that page.

Note it is an example of an inclusion map and some authors will probably call it by that name. [ilmath]\hookrightarrow[/ilmath]s may be used.

Proofs of continuity

1. Directly

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Easy, did on paper, it is easy. Haven't found reference but to be fair I think it goes on unstated

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2. As corollary to characteristic property of the subspace topology

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Easy, see page 52, corollary 3.9 in Lee's manifolds if stuck (shouldn't be stuck)

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References

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Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
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Someone must be out there. This isn't wrong