List of top Mathematics Questions on Metric Spaces and Topology

Let \((X,d)\) be a metric space. If \(S\) and \(T\) are subsets of \(X\), then which one of the options given below is NOT necessarily true? (For any \(Y\subseteq X\), \(\text{Int}(Y)\) denotes the interior of \(Y\) and \(\text{Clos}(Y)\) denotes the closure of \(Y\).)

Consider the metric spaces \((C[0,1],d_1)\) and \((C[0,1],d_\infty)\), under the metrics \(d_1(f,g)=\int_0^1 |f(x)-g(x)|\,dx\) and \(d_\infty(f,g)=\sup_{x\in[0,1]}\{|f(x)-g(x)|\}\), for \(f,g\in C[0,1]\). Then which one of the options given below is correct?

Consider the following statements:
(I) \((0,1)\) and \(\mathbb{R}\) are homeomorphic.
(II) \((0,1)\) and \((0,1]\) are homeomorphic.
(III) \((0,1)\) and \([0,1]\) are homeomorphic.
Choose the correct option.

Which one of the following options is necessarily true?

Which one of the following is not a valid metric on \(\mathbb{R}\)?