- #1
tomboi03
- 77
- 0
Let A, B, and A[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?
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?