Problem 7 - A First Course in Algebraic Topology

Question

(g) Let \(X\) be a topological space and let \(f: X \rightarrow Y\) be a surjective map. Let \(\mathscr{U}_{\mathrm{f}}\) denote the quotient topology on \(\mathrm{Y}\). Suppose that \(\mathscr{U}\) is a topology on \(\mathrm{Y}\) so that f: \(\mathrm{X} \rightarrow \mathrm{Y}\) is continuous with respect to this topology. Prove that if \(\mathrm{f}\) is a closed or an open mapping then \((\mathrm{Y}, \mathscr{U})\) is homeomorphic to \(\left(Y, \mathscr{U}_{f}\right)\). Furthermore. give examples to show that if \(\mathrm{f}\) is nejther open nor closed then \((\mathrm{Y}, \mathscr{H}) \neq\left(\mathrm{Y}, \|_{1}\right)\).

Step-by-Step Solution

Answer

If \(f\) is open or closed, \((Y, \mathscr{U})\) is homeomorphic to \((Y, \mathscr{U}_f)\). Otherwise, they may differ.

Step 1: Understanding the problem

We need to prove that if the map \(f: X \to Y\) is surjective and either open or closed, then the topology \(\mathscr{U}\) on \(Y\) makes \((Y, \mathscr{U})\) homeomorphic to \((Y, \mathscr{U}_f)\), where \(\mathscr{U}_f\) is the quotient topology regarding the map \(f\). Then, we provide examples where \(f\) is neither open nor closed.

Step 2: Definition of Quotient Topology

The quotient topology \(\mathscr{U}_f\) on \(Y\) is defined such that a set \(U \subseteq Y\) is open in \((Y, \mathscr{U}_f)\) if and only if \(f^{-1}(U)\) is open in \(X\). Because \(f\) is continuous with respect to both topologies, this condition holds with \(\mathscr{U}\).

Step 3: Proving Homeomorphism for Closed or Open Mapping

If \(f\) is an open mapping, any open set \(U\) in \(X\) gets mapped to an open set \(f(U)\) in \(Y\). Thus, \(\{V \subseteq Y : f^{-1}(V) \text{ is open in } X\}\) equals \(\mathscr{U}\), showing the topologies are the same (similarly for closed mappings). Therefore, the identity map is a homeomorphism from \((Y, \mathscr{U})\) to \((Y, \mathscr{U}_f)\).

Step 4: Example of Neither Open Nor Closed Mapping

Consider the function \(f: \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = x^2\). This map is neither open nor closed, as for example, the interval \((-1, 1)\) is open in \(\mathbb{R}\), but its image \((0, 1)\) is not open. In this case, \((Y, \mathscr{U})\) differs from \((Y, \mathscr{U}_f)\) because \(f^{-1}(V)\) does not always correspond to open sets in \(X\).

Key Concepts

Topological Space

In algebraic topology, a topological space is a set equipped with a topology, which is a collection of open sets that satisfy certain axioms. These spaces provide a framework to study continuity, convergence, and boundary structures.
A topological space consists of:

• A set, say, X.
• A collection of subsets of X, known as open sets.
• The collection of open sets satisfies these properties: the empty set and the set X itself are included, any union of open sets is also open, and any finite intersection of open sets is open.

By organizing a set with a topology, we can explore ideas like compactness, connectedness, and continuity within a structured environment. This foundational concept is crucial for understanding advanced topics like quotient topology and homeomorphisms.

Quotient Topology

Quotient topology deals with the construction of a new topology on a set based on a surjective function from one topological space to another. Let's consider a surjective mapping \( f: X \rightarrow Y \), where \( X \) is a topological space.
This concept helps in simplifying complex spaces by identifying and grouping points through the mapping.

• The quotient topology \( \mathscr{U}_f \) on the set \( Y \) is defined such that a subset \( U \subseteq Y \) is deemed open if the pre-image \( f^{-1}(U) \) is open in \( X \).
• This means the structure of the topology on \( Y \) is dictated by how the map \( f \) behaves with the open sets in \( X \).

Quotient topologies are used to understand how different topological structures correspond to each other through mappings, enabling the study of new spaces arising from known ones.

Homeomorphism

Homeomorphism is a key concept in topology that describes when two spaces are equivalent in terms of their topological properties. A homeomorphism is a bijective function \( g: X \rightarrow Y \) between two topological spaces that is continuous with a continuous inverse.
This implies that:

• Both spaces have the same 'shape' or 'type' of topology.
• There is a correspondence between open sets in \( X \) and open sets in \( Y \).
• If a space \( (Y, \mathscr{U}) \) is homeomorphic to \( (Y, \mathscr{U}_f) \), this means there's an identity map from \( Y \) with topology \( \mathscr{U} \) to \( Y \) with topology \( \mathscr{U}_f \), preserving the topological structure.

By showing that two spaces are homeomorphic, one can treat them interchangeably within topological studies, focusing on properties that do not depend on the specific nature of the elements but on how they are interconnected.

Open and Closed Mapping

In topology, understanding mappings between spaces is crucial to studying their properties. Specifically, open and closed mappings preserve certain structures within these spaces.
Open mapping:

• A map \( f: X \rightarrow Y \) is open if it maps open sets in \( X \) to open sets in \( Y \).
• This means the image of an open set under \( f \) remains open in \( Y \).

Closed mapping:

• A map \( f \) is closed if it maps closed sets in \( X \) to closed sets in \( Y \).
• This ensures the image of a closed set under \( f \) is closed in \( Y \).

These properties become particularly important when discussing whether \( Y \) with a certain topology \( \mathscr{U} \) is homeomorphic to \( Y \) with the quotient topology \( \mathscr{U}_f \). The exercise shows that if \( f \) is either an open or closed map, the spaces are homeomorphic, while this equivalence does not hold if \( f \) is neither.