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Since the disjoint union of $X_{0}= \{x_{1}, x_{2}, x_{3}\}$ and $X_{1}= \{x_{1}, x_{2}\}$ is: $ X_{0}\sqcup X_{1}= \{(x_{1},0), (x_{2},0), (x_{3},0), (x_{1},1), (x_{2},1)\}$ I have to ask; is $X_{0}$ a subset of $X_{0}\sqcup X_{1}$? I just would like to know because I can't tell if there ought to be an inclusion function.

The inclusion function is the identity map $id: x \mapsto x$ but the elements of the disjoint union are indexed...

sheafpants4
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1 Answers1

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No, formally $X_0$ is not a subset of $X_0\sqcup X_1$. But there is a natural injective function

$$i:X_0\to X_0\sqcup X_1$$ $$i(x)=(x,0)$$

and therefore many authors will treat $X_0$ as a subset of $X_0\sqcup X_1$ via the embedding above, i.e. they will replace $X_0$ with the image $i(X_0)$. This is so natural that most authors will completely forget about indexing.

freakish
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