Discussion
Let \(X\) be the space \(\mathbb{R}/\mathbb{Z}\) with the quotient topology induced from the usual topology on \(\mathbb{R}\). Consider the following statements.
I. \(X\) is compact.
II. \(X \setminus \{x\}\) is connected for any \(x \in X\).
Which one of the following is correct?
Answer: A
Both I and II are TRUE.
\(\mathbb{R}/\mathbb{Z}\) is homeomorphic to the circle \(S^{1}\) via \(t \mapsto e^{2\pi it}\). I: it is the continuous image of the compact interval [0, 1], so it is compact.
II: removing one point from a circle leaves a space homeomorphic to an open interval, which is connected. So both statements hold, and options declaring either false are wrong.
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