Question

Let \(X\) be a topological space equipped with an equivalence relation, \(\tilde{X}\) the set of equivalence classes, \(\pi: X \rightarrow \widetilde{X}\) the map taking each \(x \in X\) to its equivalence class, and \(\mathcal{T}=\left\\{U \subset \widetilde{X}: \pi^{-1}(U)\right.\) is open in \(\left.X\right\\} .\) a. \(\mathcal{J}\) is a topology on \(\widetilde{X}\). (It is called the quotient topology.) b. If \(Y\) is a topological space, \(f: \widetilde{X} \rightarrow Y\) is continuous iff \(f \circ \pi\) is continuous. c. \(\widetilde{X}\) is \(T_{1}\) iff every equivalence class is closed.

Short answer

The quotient topology \( \mathcal{T} \) is a topology. \( f: \widetilde{X} \to Y \) is continuous iff \( f \circ \pi \) is continuous. \( \widetilde{X} \) is \( T_1 \) iff each class is closed.

Step-by-step solution

Define the Quotient Topology

To show that \( \mathcal{T} \) is a topology on \( \widetilde{X} \), we follow the definition of a topology. A set \( \mathcal{T} \) is a topology on \( \widetilde{X} \) if it satisfies three properties: (1) both \( \emptyset \) and \( \widetilde{X} \) are in \( \mathcal{T} \); (2) \( \mathcal{T} \) is closed under arbitrary unions; (3) \( \mathcal{T} \) is closed under finite intersections.

Verify the Empty Set and Whole Space

The map \( \pi: X \to \widetilde{X} \) implies that \( \pi^{-1}(\emptyset) = \emptyset \), which is open in \( X \). Similarly, \( \pi^{-1}(\widetilde{X}) = X \), which is open in \( X \). Thus, both \( \emptyset \) and \( \widetilde{X} \) are in \( \mathcal{T} \).

Prove Arbitrary Unions are Open

Suppose \( \{ U_i \} \) is a collection of sets in \( \mathcal{T} \). By definition, \( \pi^{-1}(U_i) \) is open in \( X \) for each \( i \). Thus, the union \( \bigcup_i \pi^{-1}(U_i) = \pi^{-1}(\bigcup_i U_i) \) is open in \( X \), proving that \( \bigcup_i U_i \in \mathcal{T} \).

Demonstrate Finite Intersections are Open

Let \( U_1, U_2 \in \mathcal{T} \). Then \( \pi^{-1}(U_1) \) and \( \pi^{-1}(U_2) \) are open in \( X \). Consequently, their intersection \( \pi^{-1}(U_1 \cap U_2) = \pi^{-1}(U_1) \cap \pi^{-1}(U_2) \) is open in \( X \). Thus, \( U_1 \cap U_2 \in \mathcal{T} \). This confirms that \( \mathcal{T} \) is a topology.

Continuity of General Map

To prove that \( f: \widetilde{X} \rightarrow Y \) is continuous iff \( f \circ \pi \) is continuous, assume \( f \) is continuous. For any open set \( V \subset Y \), \( f^{-1}(V) \) is open in \( \widetilde{X} \), thus \( \pi^{-1}(f^{-1}(V)) = (f \circ \pi)^{-1}(V) \) is open in \( X \), making \( f \circ \pi \) continuous. Conversely, if \( f \circ \pi \) is continuous, for any \( V \subset Y \), \( (f \circ \pi)^{-1}(V) \) open implies \( \pi^{-1}(f^{-1}(V)) \) is open, ensuring \( f^{-1}(V) \) is open in \( \widetilde{X} \), proving \( f \) is continuous.

Verify \(T_1\) Spaces

To prove \( \widetilde{X} \) is \( T_1 \) iff every equivalence class is closed, consider \( \widetilde{X} \) to be \( T_1 \). Then, singleton sets are closed; specifically, for a point representing an equivalence class, the singleton set containing it is closed, implying all equivalence classes are closed. Conversely, if every class is closed, then for any class \( [x] \), its closure is the set itself, making \( \widetilde{X} \) \( T_1 \).

Key concepts

Quotient Topology

The quotient topology is a concept used to simplify the structure of a topological space by "gluing together" or identifying certain points. In more technical terms, it involves taking a topological space \( X \), partitioning it into equivalence classes, and then looking at the set of these equivalence classes. We denote this set of equivalence classes as \( \tilde{X} \). To construct a topology on \( \tilde{X} \), consider the set of all subsets \( U \subset \tilde{X} \) that satisfy \( \pi^{-1}(U) \) being open in \( X \). This collection of sets is defined as the quotient topology \( \mathcal{T} \).

• Both the entire space \( \tilde{X} \) and the empty set belong to \( \mathcal{T} \).
• It is closed under arbitrary unions of sets.
• It is closed under finite intersections of sets.

These properties ensure \( \mathcal{T} \) is indeed a topology. The presence of a quotient topology helps to analyze complex spaces by reducing them to simpler equivalent forms.

Continuous Functions

To understand continuous functions in the context of quotient topology, consider a map \( f: \widetilde{X} \rightarrow Y \), where \( Y \) is another topological space. We say that \( f \) is continuous if, for an open set \( V \) in \( Y \), the preimage \( f^{-1}(V) \) is open in \( \tilde{X} \). However, with quotient spaces, there's an additional twist: for \( f \) to be continuous, the function \( f \circ \pi \) must also be continuous.

• If \( f \circ \pi \) is continuous, so is \( f \).
• This means for any open subset \( V \cupset Y \), \( (f \circ \pi)^{-1}(V) \) being open in \( X \) implies \( f^{-1}(V) \) is open in \( \widetilde{X} \).
• Conversely, if \( f \) is continuous, then \( f \circ \pi \) naturally extends this continuity to the original set \( X \).

This relationship allows functions to preserve their continuity even when the underlying space is simplified or modified through the quotient process.

T1 Spaces

In topology, \( T_1 \) spaces are those where each single point is closed. This property holds an important implication for quotient spaces. The quotient space \( \widetilde{X} \) is \( T_1 \) if and only if every equivalence class in \( \widetilde{X} \) is closed in itself.

• This means, if you take any point from \( \widetilde{X} \), the "set" containing just this point (or its class) is a closed set.
• Closed equivalence classes ensure that if you have a sequence converging to some limit point within a class, it confines strictly within that class.
• Conversely, if \( \widetilde{X} \) satisfies the \( T_1 \) condition, then all of its singletons being closed implies the equivalence classes are already closed by definition.

Understanding \( T_1 \) spaces in quotient topology helps to navigate the nuanced behavior of sequences and limits in more abstract settings.