- #1
Extropy
- 17
- 0
Definitions:
"x is a limit point of A" = "All neighborhoods of x contain an infinite amount of points of A"
"x is a contact point of A" = "All neighborhoods of x contain at least one point of A"
"X is a centered system of closed sets" = "[tex]\cap A[/tex] is not empty, where A is any finite subset of X, and where X is a set of closed sets."
In some book of mine, in some proof of a theorem, the author implicitly asserted for some set X that "Since X is a set without limit points, X is closed."
Now, I do not really know how to prove that--in fact, I think it may be false.
For example take the space to be the set of integers. Let the open sets be any set of non-negative integers, sets of the form {a, -a} where a is any natural number, any unions of the above sets, and the empty set. Let N be the set of natural numbers. Surely N is without limit points--in fact, no element of the space is a limit point of any set since all elements of the space have a finite neighborhood. However, surely -1, which is not a an element of N, is a contact point.
Note: this is what the author stated:
"Theorem: If T is a compact space, then any infinite subset of T has at least one limit point.
Proof: Suppose T contains an infinite set X with no limit point. Then T contains a countable set [tex]X=\{x_1,x_2,...,x_n,...\}[/tex] with no limit point. But then the sets [tex]X_n=\{x_n,x_{n}_{+1} ,...\} (n=1, 2, ...)[/tex] form a centered system of closed sets in T with an empty intersection, i. e. T is not compact."
"x is a limit point of A" = "All neighborhoods of x contain an infinite amount of points of A"
"x is a contact point of A" = "All neighborhoods of x contain at least one point of A"
"X is a centered system of closed sets" = "[tex]\cap A[/tex] is not empty, where A is any finite subset of X, and where X is a set of closed sets."
In some book of mine, in some proof of a theorem, the author implicitly asserted for some set X that "Since X is a set without limit points, X is closed."
Now, I do not really know how to prove that--in fact, I think it may be false.
For example take the space to be the set of integers. Let the open sets be any set of non-negative integers, sets of the form {a, -a} where a is any natural number, any unions of the above sets, and the empty set. Let N be the set of natural numbers. Surely N is without limit points--in fact, no element of the space is a limit point of any set since all elements of the space have a finite neighborhood. However, surely -1, which is not a an element of N, is a contact point.
Note: this is what the author stated:
"Theorem: If T is a compact space, then any infinite subset of T has at least one limit point.
Proof: Suppose T contains an infinite set X with no limit point. Then T contains a countable set [tex]X=\{x_1,x_2,...,x_n,...\}[/tex] with no limit point. But then the sets [tex]X_n=\{x_n,x_{n}_{+1} ,...\} (n=1, 2, ...)[/tex] form a centered system of closed sets in T with an empty intersection, i. e. T is not compact."
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