The upside down product sign
I recently encountered the upside down product sign in an exercise. According to Wikipedia, this stands for coproduct. However, I am not sure what it means. As a specific example, here's the exercise:
It comes after 1a. I am not sure I am interpreting it correctly. Does this mean that whenever I get a set on the lower hemisphere, I have to use $f_L$ and $f_U$ when I have a set from the upper hemisphere? Usually, when I have encountered these kinds of situations, the notation has been the same as is used in defining functions piecewise.This, however, is completely new.
$\endgroup$ 21 Answer
$\begingroup$Coproducts of topological spaces are disjoint unions (as hinted at in your exercise), given the "obvious" topology. More precisely $X \sqcup Y$ can be taken to be $(X \times \{0\}) \cup (Y \times \{1\})$ topologised as a subspace of $(X \cup Y) \times \{0, 1\}$, where $\{0, 1\}$ is given the discrete topology and the product is given the product topology (so the open sets in $X \sqcup Y$ are disjoint unions of open subsets of $X$ and open subsets of $Y$).
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