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!