Question

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?

• A. \( p \) is an open map but not a closed map
• B. \( p \) is a closed map but not an open map
• C. \( p \) is a closed map as well as an open map
• D. \( p \) is neither an open map nor a closed map

Hint:

In quotient maps, the preservation of open and closed sets depends on the relationship between the topologies of the domain and codomain. When the codomain has the discrete topology, these properties are not guaranteed to hold.

Answer 1

The Correct Option is D

We are given a quotient map \( p : ([0, 1], T_1) \to \{(0, 1\}, T_2) \) where \( T_1 \) is the subspace topology induced by the Euclidean topology on \( \mathbb{R} \) and \( T_2 \) is the discrete topology on \( \{0, 1\} \). The function \( p \) is described by the characteristic function on the interval \( [\frac{1}{2}, 1] \). A quotient map is a surjective map where a set \( U \) in the codomain is open if and only if its preimage \( p^{-1}(U) \) is open in the domain. Step 1: Understanding the quotient map A quotient map has the property that it satisfies the condition that the preimage of every open set in the codomain is open in the domain. However, the properties of openness and closedness can sometimes behave differently in quotient maps, as it depends on how the topology in the domain and codomain are related. In this case, the map \( p \) takes the interval \( [0, 1] \) and maps it to \( \{0, 1\} \). The subspace topology \( T_1 \) on the interval \( [0, 1] \) is induced by the Euclidean topology on \( \mathbb{R} \), while \( T_2 \) on the set \( \{0, 1\} \) is the discrete topology, meaning every subset of \( \{0, 1\} \) is open. Step 2: Analyzing whether \( p \) is an open map To check if \( p \) is an open map, we need to see if the image of an open set in the domain is open in the codomain. Since the codomain \( \{0, 1\} \) has the discrete topology, all subsets of \( \{0, 1\} \) are open. Therefore, for \( p \) to be an open map, the image of every open set in the domain must be an open set in the codomain, which in this case always holds because of the discrete topology. However, in quotient maps, this condition can sometimes be violated because the topology on the domain may cause the image of an open set to not be open. Step 3: Analyzing whether \( p \) is a closed map Next, to check if \( p \) is a closed map, we need to see if the image of a closed set in the domain is closed in the codomain. Since the topology on \( \{0, 1\} \) is discrete, the image of any set, closed or open, will be closed by default. However, since quotient maps don't always preserve the closedness of sets (especially in non-trivial topological spaces), it is likely that in this case the image of a closed set might not be closed in the codomain. Specifically, the way the topology on the domain and the discrete topology on the codomain interact can cause the map \( p \) to fail to preserve closed sets. Step 4: Conclusion Based on the analysis above, \( p \) is neither an open map nor a closed map because quotient maps do not generally preserve the openness or closedness of sets, particularly when the domain and codomain have different topological structures. Hence, the correct answer is: \[ \boxed{D} \quad \text{\( p \) is neither an open map nor a closed map.} \]