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union of two disjoint topological spaces is a topological space? if it is not. when this statement will be right

"union of two disjoint topological spaces is a topological space"

David C. Ullrich
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Sam Sam
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  • Do you have any intuition about how you would attempt to answer this question? – kamills Dec 19 '19 at 19:30
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    Yes if the underlying sets $X,Y$ are disjoint. Then collection ${U\cup V\mid U\in\tau_X, V\in \tau_Y}$ is a topology on $X\cup Y$. – drhab Dec 19 '19 at 19:33
  • You might want to see my question here about the so-called topological sum which is used to construct a new topological space out of disjoint topological spaces: https://math.stackexchange.com/questions/3465119/what-exactly-is-a-topological-sum – Math1000 Dec 19 '19 at 19:50
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    What do you mean by the union of topological spaces? A topological space is usually defined as an ordered pair $(X,\tau)$, where $X$ is a set and $\tau$ is a collections of subsets of $X$. There is no standard meaning for the union of two ordered pairs. – Steve Kass Dec 19 '19 at 19:50
  • @kamills I mean If I have two different topological space, let we say (X,) the first one and (Y, Γ ) for another one. is if we combined the two topological spaces by union we get a new set such that the new set is topological space? – Sam Sam Dec 19 '19 at 21:21
  • @SteveKass I mean if we have two different topological space and I do union for the two of them, (combined the two sets together by union) the new set will be topological space? – Sam Sam Dec 19 '19 at 21:24
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    You can take the union of the two sets $X$ and $Y$, but to decide if it's a topological space, you have to have in mind a specific collection of subsets of $X\cup Y$ to be the open sets of the space. Note that one of your open sets has to be $X\cup Y$, so you can't just take the open sets to be the open sets of each of the original topological spaces, since $X\cup Y$ is in neither to begin with. – Steve Kass Dec 19 '19 at 22:26
  • Why is it classified into [art]? – Hanul Jeon Dec 21 '19 at 16:37

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Yes, if we have disjoint spaces $(X, \tau_X)$ and $(Y, \tau_Y)$ we can form $X \cup Y$ and give this new set the topology $$\tau:=\{U \cup V: U \in \tau_X, V \in \tau_Y\}$$

and it's rather easy to check that $\tau$ is a well-defined topology on $X \cup Y$ (and this space is often denoted $X \coprod Y$ or $X \oplus Y$, depending on your text book tradition) and gives us the so-called sum topology on $X \cup Y$. It further has the property that $X$ and $Y$ are open in $X \coprod Y$ and as a subspace of that sum (or co-product) $X$ and $Y$ keep their old topology (so $X \to X \coprod Y, x \to x$ and the similar map $Y \to X \coprod Y, y \to y$ are open embedding maps) and the sum topology is the largest topology such that those two above maps are continuous.

So the statement is right in the sense that there is a standard accepted topology on the union of two disjoint topological spaces.

Also, if both $X$ and $Y$ are both subspaces of some larger space $Z$, $X \cup Y$ is a valid subspace and thus a space in its own right too, regardless of disjointness, even.

Henno Brandsma
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