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For the inclusion map $f: \mathbb{S}^{n-1} \to \mathbb{E}^n$, find the pushout $(u_1,u_2)$ of $(f,f)$. What is this question asking exactly? Isn't the pushout the adjunction space with the inclusion maps into it? Am I supposed to prove this is a pushout somehow?
The second part asks to show that $\mathbb{E}^n {}_f\sqcup \mathbb{E}^n \simeq \mathbb{S}^n$. Can I somehow do this using the pushout? I know how to show it via the universal property of the disjoint union topology.
Well, yes, I suppose you should show that pushouts along inclusions are adjunction spaces in the category of topological spaces. (See e.g. How to show that a diagram is a pushout in the category $\text{TOP}$?)
Given that, and the fact that pushouts are unique up to isomorphism, you just need to check that $S^n$ is a pushout of $(f,f)$. That is, if you have a map from a disk (say a north hemisphere) and a map from another disk (say a south hemisphere) to another space, and if these maps agree on the common boundary (the equator), then this induces a unique map from the sphere to the other space. This follows because the sphere is obtained by gluing the two hemispheres together.
