why $ \bigcup (X \cup Y)= X\times Y $?

Question

I have some confusion in Munkres topology . My confusion is given below marked in red colour

My attempt : Here ${\bigcup}_{x\in X} T_x = \bigcup(X \times b) \cup (x \times Y)= \bigcup (X \cup Y)$

My doubt : why $ \bigcup (X \cup Y)= X\times Y $?

Essentially, the problem is that the line labelled as "My attempt" is nonsense. — , Sep 20 '20 at 08:57
In $\rm\LaTeX$ \bigcup gives you $\bigcup$ which is an operator on a family of sets; whereas $\cup$ is an operator on two sets; you wouldn't understand if I wrote $+n+k$ as $\sum n+k$, would you? — Asaf Karagila, Sep 20 '20 at 08:57
I will add a link to an older thread which contain several various proofs: Product of connected spaces. — Martin Sleziak, Sep 20 '20 at 08:58
There's an [edit] button, just so you know. Questions are not etched into marble here. — Asaf Karagila, Sep 20 '20 at 08:59

score 3 · Answer 1 · answered Sep 20 '20 at 08:52

3

The expression $ \cup (X \cup Y)$ does not make sense.

Obviously we have

$$ X \times Y = \bigcup_x \{x\} \times Y \subset \bigcup_x T_x \subset X \times Y .$$ This implies $\bigcup_x T_x = X \times Y$.

answered Sep 20 '20 at 08:52

Paul Frost

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Actually, $\union X$ typically means the union of all sets contained in $X$ – Couchy Sep 20 '20 at 11:38
@Couchy See Asaf Karagila's comment to the question. The original version was $\cup (X \cup Y)$. But even in the version $\bigcup (X \cup Y)$ - what are sets contained in $X \cup Y$? – Paul Frost Sep 20 '20 at 13:32

Henno Brandsma · Accepted Answer · 2020-09-20T09:34:02.877

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Well, let $(p,q) \in X \times Y$. Then $$(p,q) \in p \times Y \subseteq T_p \subseteq \bigcup_{x \in X} T_x$$ As all $T_x \subseteq X \times Y$ the other inclusion is trivial.

edited Sep 20 '20 at 09:34

answered Sep 20 '20 at 08:53

Henno Brandsma

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..@Henno sir i thinks it should be $(p,q) \in p \times Y$ – jasmine Sep 20 '20 at 09:21

why $ \bigcup (X \cup Y)= X\times Y $?

2 Answers2