Problem 1
Let \(X\) be the set of all ordinal numbers less than the first uncountable
ordinal. Let \((\alpha, \beta) \subset X\) denote the set of all ordinal numbers
\(\gamma\) such that \(\alpha
Problem 2
A topological space \(T\) is said to be locally compact if every point \(x \in T\) has at least one relatively compact neighborhood. Show that a compact space is automatically locally compact, but not conversely. Prove thatevery closed subspace of a locally compact subspace is locally compact.
Problem 3
\(A\) point \(x\) is said to be a complete limit point of a subset A of a topological space if, given any neighborhood \(U\) of \(x\), the sets \(A\) and \(A \cap U\) have the same power (i.e., cardinal number). Prove that every infinite subset of a compact topological space has at least one complete limit point.
