Discussion
Let S' denote the unit circle with centre origin which of the following space is not connected ?
Answer: C
The disconnected space is S' \(\setminus \{P_{1}, P_{2}\} \times\) [0, 1].
Removing two distinct points from a circle leaves two disjoint open arcs. A product with the connected interval [0, 1] keeps the two pieces apart, so the space is the union of two disjoint non-empty open sets and is not connected.
The others are connected. S' × [0, 1] is a closed annulus (a cylinder), a product of connected spaces. Removing one point from the circle leaves a single open arc, homeomorphic to an interval, so S' \(\setminus\) {P} × [0, 1] is connected. S' × {0} is just a copy of the circle.
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