I want to show that the union $\{0\}\cup(0,1)=[0,1)$ is not a disjoint union of topological spaces. It's my first time handling such a problem and I don't even know when is a topological space a disjoint union or not. I tried referring to the Universal Property of disjoint union but couldn't see how it helps me. Any hint would help.
2026-03-27 16:53:52.1774630432
Disjoint union of topological spaces
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Hint: In the disjoint union topology on $[0, 1)$ (using your sets), $\{0\}$ would be open. Can you prove that it is not open in the usual topology on $[0, 1)$? Moreover, could you prove that no singleton is open in $[0,1)$?