- #1
Bipolarity
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In the textbook I am working with, an isolated point of A is defined to be a point X in A such that there exists a neighborhood (open ε-ball) centered on X containing no point in A other than X itself.
A boundary point of A (which need not be in A) is defined as a point X in A such that every open ε-ball centered on X contains at least one point in A and at least one point not in A.
Is it the case that isolated points must be boundary points? Most textbooks use definitions other than the one I'm using, hence I am very confused.
Also, maybe this is true for metric spaces but not for topological spaces?
BiP
A boundary point of A (which need not be in A) is defined as a point X in A such that every open ε-ball centered on X contains at least one point in A and at least one point not in A.
Is it the case that isolated points must be boundary points? Most textbooks use definitions other than the one I'm using, hence I am very confused.
Also, maybe this is true for metric spaces but not for topological spaces?
BiP