Problem 3
Consider the set \(\mathscr{T}\) of all possible topologies defined in a set \(X\), where \(\tau_{2} \leqslant_{0} \tau_{1}\) means that \(\tau_{2}\) is weaker than \(\tau_{1}\). Verify that \(\leqslant\) is a partial ordering of \(\mathscr{T}\). Does \(\mathscr{T}\) have maximal and minimal elements? If so, what are they?
Problem 4
Can two distinct topologies \(\tau_{1}\) and \(\tau_{2}\) in X generate the same relative topology in a subset \(A \subset X\) ?
Problem 5
Let $$ X=\\{a, b, c\\}, \quad A=\\{a, b\\}, \quad B=\\{b, c\\} $$ and let \(\mathscr{G}=\\{\varnothing, X, A, B) .\) Is \(\mathscr{G}\) a base for a topology in \(\mathrm{X}\) ?
Problem 6
Prove that if \(M\) is an uncountable subset of a topological space with a countable base, then some point of \(M\) is a limit point of \(M\).
Problem 8
Prove that a topological space satisfying the second axiom of countability automatically satisfies the first axiom of countability.
Problem 9
Give an example of a topological space satisfying the first axiom of countability but not the second axiom of countability.
Problem 10
Let \(\tau\) be the system of sets consisting of the empty set and every subset of the closed unit interval \([0,1]\) obtained by deleting a finite or countable number of points from \(X\). Verify that \(T=(X, \tau)\) is a topological space. Prove that \(\mathrm{T}\) satisfies neither the second nor the first axiom of countability. Prove that \(\mathrm{T}\) is a \(T_{1}\)-space, but not a Hausdorff space.
