# Union of sets | quick question

1. Jun 3, 2010

### 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

2. Jun 3, 2010

### Dick

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

3. Jun 4, 2010

### HallsofIvy

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

4. Jun 4, 2010

### michonamona

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?

5. Jun 4, 2010

Correct

6. Jun 4, 2010

Thanks!