1.2 Continuous functions and homeomorphisms
3 min read•august 9, 2024
Continuous functions and homeomorphisms form the backbone of topological relationships. These concepts allow us to compare and classify spaces based on their fundamental properties, bridging the gap between geometry and algebra in topology.
Understanding these ideas is crucial for grasping how topological spaces relate to each other. We'll explore how continuous maps preserve closeness, and how homeomorphisms establish , setting the stage for deeper topological investigations.
Continuous Functions and Maps
Understanding Continuous Functions
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- preserves closeness between points in topological spaces
- Maps open sets in domain to open sets in codomain
- Characterized by inverse images of open sets remaining open
- Epsilon-delta definition applies in metric spaces
- Intuitively represents functions without "breaks" or "jumps"
- Examples include polynomials, exponential functions, and trigonometric functions
- Counterexample includes step functions (discontinuous at jump points)
Types of Topological Maps
- Open map sends open sets to open sets in the codomain
- Preserves openness of sets under the function
- Not all continuous functions are open maps
- Closed map sends closed sets to closed sets in the codomain
- Preserves closedness of sets under the function
- Some functions can be both open and closed maps
- Embedding represents a homeomorphism between a space and its image
- Preserves topological properties of the original space
- Allows study of spaces as subspaces of simpler or more familiar spaces
Homeomorphisms and Topological Invariants
Exploring Homeomorphisms
- Homeomorphism establishes topological equivalence between spaces
- Bijective continuous function with continuous inverse
- Preserves all topological properties between spaces
- Examples include stretching, bending, or twisting without tearing or gluing
- Sphere and cube are homeomorphic (can be deformed into each other)
- Torus and coffee mug are homeomorphic (classic topological joke)
- Counterexample includes sphere and torus (different number of holes)
Identifying Topological Invariants
- Topological invariant remains unchanged under homeomorphisms
- Used to distinguish non-
- Examples include dimension, compactness, connectedness, and fundamental group
- Euler characteristic serves as an invariant for surfaces
- Betti numbers provide information about the number of holes in different dimensions
- Homotopy groups generalize the fundamental group to higher dimensions
- Homology groups offer algebraic tools to study topological spaces
Exploring Retractions
- maps a space onto a subspace while fixing points in the subspace
- Continuous function satisfying for all x in the domain
- Image of retraction is called a retract of the original space
- Examples include projecting a filled disk onto its boundary circle
- Deformation retraction continuously deforms a space onto a subspace
- Used to simplify topological spaces while preserving essential features
- Important in homotopy theory and the study of CW complexes
Homotopy
Understanding Homotopy and Its Applications
- Homotopy represents a continuous deformation between two continuous functions
- Provides a way to classify maps up to continuous deformation
- Homotopy equivalence generalizes homeomorphism
- Path homotopy relates to the fundamental group of a space
- Homotopy groups extend the concept to higher dimensions
- Used in algebraic topology to study topological spaces
- Examples include contracting a loop in a simply connected space
- Counterexample includes loops around a hole in a torus (not contractible)
- Homotopy type captures essential topological features of a space
- CW complexes provide a framework for studying spaces up to homotopy equivalence
- Cellular homology uses homotopy to compute homology groups efficiently
Key Terms to Review (15)
Basis: In topology, a basis is a collection of open sets in a topological space such that every open set can be expressed as a union of these basis sets. The concept of a basis is essential because it allows us to define the structure of the topology in a more manageable way. By using a basis, we can analyze properties like continuity and homeomorphisms through the lens of simpler open sets.
Bijection: A bijection is a type of function that establishes a one-to-one correspondence between two sets, meaning every element in the first set is paired with exactly one unique element in the second set, and vice versa. This characteristic makes bijections particularly important when discussing concepts like continuity and homeomorphisms, as they ensure that the two sets retain their structural similarities during mappings.
Closed mapping: A closed mapping is a type of function between topological spaces where the image of every closed set under the mapping is also closed in the codomain. This concept is important in understanding how different spaces relate to each other, especially when discussing properties like continuity and homeomorphisms. Closed mappings help in analyzing how the structure of a space can be preserved when transformed, which is fundamental in topology.
Continuous function: A continuous function is a mathematical function where small changes in the input result in small changes in the output, meaning that the function does not have any jumps, breaks, or holes. This property ensures that the graph of the function can be drawn without lifting your pencil off the paper. In topology, continuous functions are crucial because they help to understand how spaces behave and relate to one another, especially in concepts like homeomorphisms.
Deformation retract: A deformation retract is a type of homotopy between two continuous functions where one space can be continuously shrunk to another, effectively preserving its topological properties. This concept is crucial in understanding the idea of spaces being 'the same' from a topological perspective, as it helps establish equivalence between spaces through continuous maps. In essence, deformation retracts provide a way to simplify complex spaces while maintaining their essential features.
Homeomorphic Spaces: Homeomorphic spaces are topological spaces that can be transformed into one another through continuous functions with continuous inverses. This means there exists a bijective function between the two spaces that preserves the structure of open sets, ensuring that the spaces share the same topological properties, like connectedness and compactness.
Open Mapping: An open mapping is a function between two topological spaces that takes open sets to open sets. This property is crucial in understanding how functions behave, especially in the context of continuous transformations and inverses. Open mappings indicate that the function preserves the 'openness' of sets, which is important for ensuring that local properties around points in one space correspond well to local properties in another space.
Retraction: Retraction is a continuous mapping from a topological space into a subspace that leaves points of the subspace fixed. This concept is crucial in understanding how spaces relate to one another, as it helps illustrate how one space can 'retract' to a simpler or smaller part while maintaining some topological properties. It often appears in discussions about continuous functions and homeomorphisms, emphasizing how certain properties can be preserved under these mappings.
Subspace topology: Subspace topology is a way to create a new topological space from an existing one by restricting the open sets of the larger space to a subset. This new topology on the subset consists of intersections of the open sets of the original space with the subset, allowing us to retain the topological properties while focusing on a smaller set. This concept is essential when understanding how continuous functions and homeomorphisms behave between different spaces and how topological properties can be inherited or altered when working with subsets.
The line with two origins: The line with two origins is a classic example in topology that illustrates the concept of non-homeomorphic spaces. It is created by taking two copies of the real line and identifying them at all points except one, resulting in a space that has two distinct origins. This example shows how continuous functions can behave differently based on the topological properties of the spaces involved, specifically demonstrating how homeomorphisms preserve certain characteristics.
The topologist's sine curve: The topologist's sine curve is a classic example in topology that illustrates how certain subsets of the Euclidean space can exhibit properties that differ from the full space. It consists of the set of points given by the graph of the function $y = ext{sin}(1/x)$ for $x > 0$, along with the segment on the line $x = 0$ for $y$ in the interval $[-1, 1]$. This construction highlights concepts like continuity and convergence within the context of homeomorphisms, as well as the behavior of functions defined on compact spaces.
Tietze Extension Theorem: The Tietze Extension Theorem states that if you have a normal topological space and a closed subset within it, then any continuous function defined on that closed subset can be extended to a continuous function defined on the entire space. This theorem is crucial for understanding how continuous functions behave in relation to compactness and connectedness, and it provides a foundational tool for working with partitions of unity.
Topological equivalence: Topological equivalence is a concept that describes when two topological spaces can be considered the same from a topological standpoint, meaning they can be transformed into one another through continuous functions. This transformation must be a homeomorphism, which is a continuous function that has a continuous inverse. Such equivalence captures the idea that topological properties, like connectedness and compactness, are preserved under these transformations.
Topological Space: A topological space is a set of points, along with a collection of open sets that satisfy specific properties, allowing us to define concepts such as continuity, convergence, and connectedness. It provides the foundational framework for many areas in mathematics, enabling us to study geometric properties in an abstract way. The structure of a topological space allows us to explore how different mathematical objects relate to each other through continuous transformations and mappings.
Urysohn's Lemma: Urysohn's Lemma states that if a space is normal, then for any two disjoint closed sets in that space, there exists a continuous function that maps the space to the interval [0, 1] such that it takes the value 0 on one closed set and 1 on the other. This concept connects deeply with continuous functions, showing how one can create mappings that respect the topology of a space. The lemma is essential for understanding separation properties and plays a critical role in the study of compactness.