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 $?

Answer 1

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.

Answer 2

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 $\bigcup_x T_x = X \times Y$.