- What is an interior point? An isolated point? A limit point?
Solution
1. Let \(X\) be a topological space and \(A \subseteq X\).
setup.
2. A point \(x \in A\) is an interior point of \(A\) if there is an open set \(U\) with \(x \in U \subseteq A\).
definition of interior point.
3. A point \(x \in A\) is an isolated point of \(A\) if there is an open set \(U\) with \(U \cap A = \{x\}\).
definition of isolated point.
4. A point \(x \in X\) is a limit point (accumulation point) of \(A\) if every open set \(U\) containing \(x\) meets \(A\) in a point other than \(x\), i.e. \((U \setminus \{x\}) \cap A \neq \varnothing\).
definition of limit point.
5. Q.E.D.
2, 3, 4.