Is the Union of Open Sets Also Open in Y?

[michonamona]

1. Suppose open sets $$V_{\alpha}$$ where $$V_{\alpha} \subset Y \: \forall \alpha$$, is it true that the union of all the $$V_{\alpha}$$ will belong in Y? (i.e. $$\bigcup_{\alpha} V_{\alpha} \subset Y$$)

Thanks!
M

 

[Dick]

Of course it's true. If you aren't sure, I think you'd better try and prove it.

 

[HallsofIvy]

Let x be an element of that union. Then what must be true about x?

 

[michonamona]

Let x be an element of that union. Then what must be true about x?

Ok, if x is a member of $$\bigcup_{\alpha} V_{\alpha}$$ then x is a member of $$V_{\alpha}$$ for some $$\alpha$$. But $$V_{\alpha} \subset Y \: \forall \alpha$$. Then x is also an element of Y. Since this is true for every x in $$\bigcup_{\alpha} V_{\alpha}$$, then it must be the case that $$\bigcup_{\alpha} V_{\alpha} \subset Y \: \forall \alpha$$.

Was that convincing?

 

[VeeEight]

Correct

 

[michonamona]

Thanks!