Discussion
Let \(p : ([0,1], T_{1}) \to (\{0,1\}, T_{2})\) be the quotient map, arising from the characteristic function on \([\frac{1}{2}, 1]\), where \(T_{1}\) is the subspace topology induced by the Euclidean topology on \(\mathbb{R}\). Which of the following statements is TRUE?
Answer: D
The map p is neither an open map nor a closed map.
p sends [0, 1/2) to 0 and [1/2, 1] to 1. In the quotient topology a set is open when its preimage is open in [0, 1]. The preimage of {0} is [0, 1/2), open in [0, 1]; the preimage of {1} is [1/2, 1], not open. So \(T_{2}\) = {∅, {0}, {0, 1}}, whose closed sets are {0, 1}, {1} and ∅.
Not open: (1/2, 1] is open in [0, 1] but its image {1} is not open. Not closed: [0, 1/4] is closed in [0, 1] but its image {0} is not closed.
So the options claiming p is open, closed, or both are all wrong.
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