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Isolated point

  • Thread starter kathrynag
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Homework Statement


Let a be an element of A. Prove that A is an isolated point of A iff there exists an epsilon neighborhood V(a) such that V(a)[tex]\cap[/tex]A={a}



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The Attempt at a Solution


A point is an isolated point if it is not a limit point.
Let a be an element of A.
Let be an isolated point. We want to show V(a)[tex]\cap[/tex]A={a}.
Since a is not a limit point, we say x=lim[tex]a_{n}[/tex] satisfying [tex]a_{n}[/tex]=x
 

Answers and Replies

  • #2
jbunniii
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Try proving this equivalent statement:

A is a limit point iff there does not exist an epsilon neighborhood [itex]V(a)[/itex] such that [itex]V(a) \cap A = \{a\}[/itex].

Notice that the following mean the same thing:

"there does not exist an epsilon neighborhood [itex]V(a)[/itex] such that [itex]V(a) \cap A = \{a\}[/itex]"

"every epsilon neighborhood [itex]V(a)[/itex] contains at least one point of [itex]A[/itex] that is distinct from [itex]a[/itex]."
 

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