Is the Union of Open Sets Also Open in Y?

[michonamona]

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

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

Was that convincing?

 

[VeeEight]

Correct

 

[michonamona]

Thanks!