Homework 5: Limit Points

Instructions. Assignments should be stapled and written clearly and legibly.

1. Section 2.2, #2.13(a)(c)(e)(f), 2.15.
2. Let \(X\) be a topological space and \(A\subseteq X\).
   1. Prove: if a sequence \((x_1,x_2,\ldots)\) of points in \(A\) converges to \(x\), then \(x\in \overline{A}\).
   2. Prove: if a sequence \((x_1,x_2,\ldots)\) of points in \(A\setminus\{x\}\) converges to \(x\), then \(x\in A'\).
3. A topological space \(X\) is said to be a \(T_1\)-space if finite subsets of \(X\) are closed. Prove that a Hausdorff space is a \(T_1\)-space.
   Hint. Use a theorem we proved in class about Hausdorff spaces.
4. Suppose that a topological space \(X\) is a \(T_1\)-space. Let \(A\subseteq X\).
   1. Prove that \(x\) is a limit point of \(A\) if and only if every neighborhood of \(x\) intersects \(A\) in infinitely many points.
   2. Prove that \((A')'\subseteq A'\).