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_{[tex]\alpha[/tex]}denote subsets of a space X.

neighborhood of [tex]\bigcup[/tex]A

_{[tex]\alpha[/tex]}[tex]\supset[/tex] [tex]\bigcup[/tex] neighborhood of A

_{[tex]\alpha[/tex]}; give an example where equality fails.Criticize the following "proof" of the above statement: if {A

_{[tex]\alpha[/tex]}} is a collection of sets in X and if x [tex]\in[/tex] neighborhood of [tex]\bigcup[/tex]A

_{[tex]\alpha[/tex]}, then every neighborhood U of x intersects [tex]\bigcup[/tex] A

_{[tex]\alpha[/tex]}. Thus U must intersect some A

_{[tex]\alpha[/tex]}, so that x must belong to the closure of some A

_{[tex]\alpha[/tex]}. Therfore, x [tex]\in[/tex] [tex]\bigcup[/tex] neighborhood of A

_{[tex]\alpha[/tex]}.

I know that the error is in the statement "x must belong to some A

_{[tex]\alpha[/tex]}closure over the whole thing." it is false...

but I don't know how to explain it...

Can you help me out guys?