Problem 16 - A First Course in Algebraic Topology

Question

(p) Let \(\mathrm{p}: \tilde{\mathrm{X}} \rightarrow \mathrm{X}\) be a covering and let \(\mathrm{f}, \mathrm{g}: \mathrm{Y} \rightarrow \widetilde{\mathrm{X}}\) be two continuous maps with \(\mathrm{pf}=\mathrm{pg}\). Prove that the set of points in \(\mathrm{Y}\) for which \(\mathrm{f}\) and \(\mathrm{g}\) agree is an open and closeá subset of \(\mathrm{Y}\).

Step-by-Step Solution

Answer

Set \( A \) is both open and closed in \( \mathrm{Y} \), implying it's the entire space.

Step 1: Understanding the Covering Map

The function \( p: \tilde{\mathrm{X}} \rightarrow \mathrm{X} \) is a covering map, which implies it is a continuous surjective map where each point in \( \mathrm{X} \) has a neighborhood evenly covered by \( \tilde{\mathrm{X}} \). This property will be pivotal in our proof.

Step 2: Defining the Set of Agreement

Define the set of points where \( f \) and \( g \) agree as \( A = \{ y \in \mathrm{Y} \mid f(y) = g(y) \} \). Our goal is to show that \( A \) is both open and closed in \( \mathrm{Y} \).

Step 3: Showing A is Open

Consider any point \( y_0 \in A \). By definition, we have \( f(y_0) = g(y_0) \). Since \( f(y_0) = g(y_0) \) projects to the same point in \( \mathrm{X} \), there is a neighborhood \( V \) in \( \mathrm{X} \) that is evenly covered by \( \tilde{\mathrm{X}} \) such that \( p^{-1}(V) \) contains disjoint open subsets covering \( V \). Choose \( U \) in \( \mathrm{Y} \) such that \( f(U) \subseteq p^{-1}(V) \) which will imply \( g(U) \subseteq p^{-1}(V) \) as well due to \( pf = pg \). If \( f(y) eq g(y) \) for some \( y \in U \), \( p(f(y)) eq p(g(y)) \) contradicting \( pf = pg \), hence \( U \subseteq A \), making \( A \) open.

Step 4: Showing A is Closed

For \( A \) to be closed, its complement \( Y - A \) must be open. Suppose \( y_0 \in Y - A \), meaning \( f(y_0) eq g(y_0) \). Again, consider a simple neighborhood in \( X \) around \( p(f(y_0)) = p(g(y_0)) \), and because \( y_0 \in Y - A \), all paths or lifts of paths around \( y_0 \) would not maintain equality, thus these points remain in the complement of \( A \), proving that \( Y - A \) is open. Consequently, \( A \) is closed.

Step 5: Conclusion on Set A

Since set \( A \) is both open and closed in a connected space \( \mathrm{Y} \), and contains at least one point (if it contains a neighborhood), by the connectedness of \( \mathrm{Y} \), it must be the entire space or empty. Given the shared properties of \( f \) and \( g \) due to \( pf = pg \), the disagreement open subset cannot exist, thus it is the entire space if nonempty.

Key Concepts

Continuous Maps

Continuous maps are fundamental in topology and mathematics broadly.
They are mappings between two topological spaces that preserve the notion of closeness.
This means that if a point maps to another point under a continuous function, nearby points will map to nearby points.

To be more precise, a map \( f: X \to Y \) is continuous if for every open set \( V \) in \( Y \), the preimage \( f^{-1}(V) \) is an open set in \( X \).
This property ensures that the mapping does not "tear or smash" the space, which is crucial when working with concepts like covering maps.

• Continuous maps maintain the structure of the space, ensuring the pattern of connection among points stays consistent.
• They are the backbone behind the notion of homeomorphisms, which capture when two topological spaces are "essentially the same".

Continuous maps also play a pivotal role in proving topological properties, such as when arguing about the openness or closedness of certain sets via covering maps.

Open and Closed Sets

Understanding open and closed sets is central to mastering topology.
Open sets are collections of points in a topological space that intuitively do not include their boundary.
For example, the open interval \((a, b)\) does not contain the endpoints \(a\) and \(b\).

Closed sets complement open sets.
A set is closed if its complement with respect to the entire space is open.
In the real numbers, the closed interval \([a, b]\) includes the endpoints.

• Every space has open sets, but not all sets in a space are open.
• Both open and closed sets can be critical when discussing the topology of spaces as they describe the shape and structure.

In proof strategies like the one presented, determining whether a set is open or closed helps in discussing the properties of spaces and maps, such as showing a specific set is both, indicating a stronger property such as in a connected space.

Connected Spaces

Connected spaces are an exciting concept in topology.
A space is called connected if it cannot be divided into two disjoint non-empty open subsets.
Intuitively, this means there's no break or hole that splits the space into separate parts.

The significance of connected spaces comes especially when discussing continuity and covering spaces.
In connected spaces, certain properties hold everywhere if they hold somewhere, giving these spaces a sense of unity.

• Connected spaces reinforce the idea that a property like openness or closedness applies globally once identified locally.
• This characteristic is leveraged in proofs to conclude about the whole space from its subparts.

For example, if a set in a connected space is both open and closed, it must either be empty or the entire space itself, which makes connected spaces ideal for such logical deductions.

Neighborhoods in Topology

Neighborhoods in topology offer a lens into the local properties of spaces.
A neighborhood of a point is a set that includes an open set containing the point itself.
This concept allows us to zoom into the behavior of spaces near specific points and plays a crucial role in describing continuity and other topological properties.

In our context, neighborhoods help in the examination of continuous maps, particularly in covering spaces, where the neighborhood properties guide us in understanding maps' behaviors.

• Neighborhoods demonstrate how spaces can locally resemble each other, despite potentially different global structures.
• They aid in establishing other concepts too, such as compactness and local connectedness.

By using neighborhoods around similar points, we can demonstrate properties like open intersections and analyze how coverings like \( p: \tilde{X} \to X \) function over these domains.