2.1 Continuous functions and their properties
4 min read•july 30, 2024
Continuous functions are the backbone of topology, bridging the gap between different spaces. They preserve key properties like and , allowing us to transfer information from one space to another.
Understanding continuity in topological spaces is crucial for grasping homeomorphisms. These special continuous functions with continuous inverses form the foundation for topological equivalence, helping us classify spaces and study their fundamental properties.
Continuity for Topological Spaces
Defining Continuity in Abstract Settings
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- Continuity in topological spaces generalizes continuity from real analysis to more abstract settings
- Function f: X → Y between topological spaces X and Y becomes continuous when preimage of every open set in Y remains open in X
- Equivalent continuity definitions involve
- Preimage of every closed set staying closed
- Image of every becoming limit point of the image
- Neighborhoods play crucial role in defining continuity for topological spaces
- or generalize concept of sequences in metric spaces to characterize continuity
- at a point utilizes open neighborhoods in domain and codomain
Advanced Continuity Concepts
- outlines conditions for combining functions continuous on subspaces to form on entire space
- Continuity characterization using nets or filters extends beyond metric spaces
- Local vs. distinctions arise in topological spaces
- concept generalizes to uniform spaces
- and provide weaker notions of continuity in topological spaces
Properties of Continuous Functions
Preservation Properties
- Connected set image under continuous function remains connected (preservation of connectedness)
- Example: continuous function mapping interval [0,1] to a circle preserves connectedness
- Compact set image under continuous function stays compact (preservation of compactness)
- Example: continuous function from closed interval [0,1] to real line maps to compact subset
- Continuous functions preserve convergence of nets and filters
- Example: convergent sequence in domain maps to convergent sequence in codomain
- Continuous functions preserve between topological spaces
- Example: continuous between preserves Hausdorff property
Operational Properties
- Composition of continuous functions yields continuous function (closure under composition)
- Example: f(x) = sin(x) and g(x) = x^2 are continuous, so g(f(x)) = sin^2(x) is continuous
- Restriction of continuous function to subspace remains continuous
- Example: sine function restricted to [0, π] is continuous
- Product of continuous functions is continuous in
- Example: f(x,y) = (sin(x), cos(y)) is continuous as a function from R^2 to R^2
Constructing Continuous Functions
Basic Continuous Functions
- Constant functions between topological spaces always maintain continuity
- Example: f(x) = 5 for all x in R is continuous
- Identity functions on topological spaces preserve continuity
- Example: f(x) = x on any is continuous
- in product spaces retain continuity
- Example: π_1(x,y) = x from R^2 to R is continuous
- induce continuous functions on quotient spaces
- Example: projection from sphere to real projective plane is continuous
Advanced Construction Techniques
- provide method to continuously deform one continuous function into another
- Example: linear homotopy between f(x) = x and g(x) = x^2 given by H(x,t) = (1-t)x + tx^2
- allows construction of continuous functions by combining continuous functions defined on closed subsets
- Example: constructing a continuous function on [-1,1] by gluing functions defined on [-1,0] and [0,1]
- Extension theorems () provide conditions for extending continuous functions from subspaces to entire space
- Example: extending a continuous function from a closed subset of a normal space to the whole space
Continuous Functions: Composition and Restriction
Composition Properties
- Composition of two continuous functions yields continuous function (closure under composition)
- Right cancellation property states if f ∘ g is continuous and f is injective, then g is continuous
- Left cancellation property asserts if f ∘ g is continuous and f is surjective, then f is continuous
- Continuity preserves under pullbacks and pushouts in category theory
- Example: pullback of continuous functions in topological spaces remains continuous
Restriction and Product Properties
- Restriction of continuous function to subspace maintains continuity
- Example: exponential function restricted to positive real numbers remains continuous
- Product of continuous functions stays continuous in product topology
- Example: f(x,y) = (e^x, ln(y)) is continuous as a function from R^2 to R^2
- Local continuity does not always imply global continuity, but holds true for functions between first-countable spaces
- Example: function continuous at each rational point may not be continuous everywhere
Key Terms to Review (34)
Bijection: A bijection is a type of function that establishes a one-to-one correspondence between the elements of two sets, meaning every element in the first set is paired with exactly one unique element in the second set, and vice versa. This concept is crucial for understanding the properties of continuous functions, as it ensures that the function has an inverse that is also a function, which plays a key role in various topological considerations.
Bolzano-Weierstrass Theorem: The Bolzano-Weierstrass Theorem states that every bounded sequence in \\mathbb{R}^n has a convergent subsequence. This important result highlights the connection between boundedness and compactness, revealing how these concepts relate to continuous functions and their properties, particularly in the context of limits and convergence.
Cliquishness: Cliquishness refers to the tendency of individuals or groups to form exclusive relationships, creating tightly-knit circles that can be resistant to outsiders. This behavior often leads to a lack of openness in communication and can hinder collaboration across different groups. In the context of continuous functions and their properties, cliquishness can be understood as a metaphor for how certain properties or functions might behave in isolation rather than in relation to broader topological spaces.
Closure of a set: The closure of a set refers to the smallest closed set that contains the original set, including all its limit points. This concept is essential in understanding how sets behave under continuous functions and helps in analyzing the properties of convergence and compactness in topology.
Compactness: Compactness is a property of a topological space that essentially means it can be covered by a finite number of open sets. This property connects to various important features, such as continuity, convergence, and the behavior of functions defined on these spaces. Compact spaces are crucial in various contexts, as they often allow for nice results like every open cover having a finite subcover and the continuity of functions leading to compact images.
Composed Functions: Composed functions are the result of combining two or more functions, where the output of one function becomes the input of another. This process is fundamental in understanding how different functions interact and behave, especially when exploring properties like continuity and limits. Composing functions helps in analyzing complex systems by breaking them down into simpler components that can be studied individually.
Connectedness: Connectedness refers to a property of a topological space where it cannot be divided into two or more disjoint non-empty open sets. This concept highlights how elements within a space relate to one another and can be crucial for understanding the overall structure and behavior of the space in question. It plays a significant role in examining continuity, the nature of certain functions, and the interactions between different mathematical objects.
Constant function: A constant function is a type of mathematical function that assigns the same output value for every input value. This means that no matter what input you provide, the output will remain unchanged, illustrating a horizontal line on a graph. Constant functions play a key role in understanding continuous functions as they exemplify the simplest case of continuity.
Continuous Function: A continuous function is a mapping between two topological spaces where the pre-image of every open set is open. This means that small changes in the input result in small changes in the output, maintaining the overall structure and behavior of the space. Continuous functions preserve limits and can be analyzed within various contexts, including subspaces, fixed points, homeomorphisms, and their inherent properties.
Filters: In mathematics, particularly in topology, a filter is a collection of subsets of a given set that allows us to formalize the idea of convergence and limit points in a structured way. Filters help in defining continuity and limits of functions by providing a way to approach points in a topological space while capturing the notion of 'largeness' and 'eventuality' of sets in relation to other sets. They play a crucial role in characterizing continuous functions and their properties.
Global Continuity: Global continuity refers to the property of a function being continuous over its entire domain, ensuring that small changes in the input lead to small changes in the output. This concept emphasizes the uniform behavior of the function across its entire range, rather than just in localized neighborhoods. Understanding global continuity is essential because it helps identify whether a function maintains its continuity across all points without any breaks or jumps.
Gluing Lemma: The gluing lemma is a fundamental concept in topology that states if two continuous functions agree on a shared subset, then these functions can be 'glued' together to create a new continuous function on a larger space. This idea is crucial because it ensures that we can extend our continuous functions without losing the property of continuity, especially when working with open sets or topological spaces.
Hausdorff Spaces: A Hausdorff space is a topological space where for any two distinct points, there exist disjoint neighborhoods around each point. This property ensures that points can be 'separated' in a way that reflects a strong sense of distinctness. The Hausdorff condition is crucial in many areas of topology, particularly in relation to convergence and continuity, as it guarantees that limits of sequences (if they exist) are unique.
Heine-Cantor Theorem: The Heine-Cantor Theorem states that if a function is continuous on a compact metric space, then it is uniformly continuous on that space. This theorem connects the concepts of continuity and compactness, indicating that continuity alone is not enough to ensure uniformity in the behavior of a function over its entire domain.
Homeomorphism: A homeomorphism is a continuous function between two topological spaces that has a continuous inverse, establishing a one-to-one correspondence that preserves the topological structure. This means that two spaces are considered homeomorphic if they can be transformed into each other through stretching, bending, or twisting, without tearing or gluing. Homeomorphisms are fundamental in determining when two spaces can be regarded as essentially the same in a topological sense.
Homotopies: Homotopies are continuous deformations between two continuous functions defined on the same topological space. This means that if you have two functions, a homotopy shows how you can 'morph' one function into another without breaking or tearing the space, maintaining continuity throughout the transformation. Homotopies play a key role in understanding the properties of spaces and maps in algebraic topology, particularly in classifying spaces based on their fundamental characteristics.
Identity function: An identity function is a special type of function that always returns the same value that was used as its input. This means for any element 'x' in the set, the identity function maps 'x' to itself, effectively leaving it unchanged. This concept is crucial in understanding how functions can behave under composition and serves as a foundational example when discussing continuous functions and their properties.
Image of a function: The image of a function refers to the set of all output values that a function can produce based on its input values. This concept is crucial in understanding how functions map elements from one set, called the domain, to another set, known as the codomain. The image can reveal important properties of the function, such as whether it is onto (surjective) or not, and it helps in analyzing the behavior of continuous functions in various contexts.
Intermediate Value Property: The intermediate value property states that if a function is continuous on a closed interval and takes on two different values at the endpoints of that interval, then it must take on every value between those two values at least once within that interval. This property highlights a fundamental characteristic of continuous functions, emphasizing their unbroken nature and the concept of 'no jumps' in their outputs.
Isomorphism: An isomorphism is a structure-preserving map between two mathematical objects that demonstrates a one-to-one correspondence, ensuring that their underlying structures are essentially the same. This concept allows us to identify when different mathematical representations or structures are fundamentally equivalent, which is crucial in various areas such as algebra, topology, and category theory.
Limit point: A limit point of a set is a point where every neighborhood of that point contains at least one point from the set, distinct from the point itself. This concept highlights how points can cluster around others, and it's essential for understanding continuity and the behavior of functions, as well as distinguishing between open and closed sets.
Local continuity: Local continuity refers to the property of a function being continuous at a specific point in its domain. This means that for any point in the domain, the function behaves predictably in the neighborhood around that point, allowing for small changes in input to result in small changes in output. It is an essential aspect of understanding continuous functions, as it helps in exploring how functions behave near particular points and provides a foundation for further analysis like limits and differentiability.
Metric Space: A metric space is a set equipped with a function, called a metric, that defines a distance between any two elements in the set. This structure allows for the formalization of concepts like convergence, continuity, and compactness. Understanding metric spaces is crucial for discussing fixed point theorems and the properties of continuous functions, as they provide a foundational framework to analyze these mathematical ideas in terms of distances and neighborhoods.
Nets: In topology, a net is a generalization of the concept of sequences used to study the convergence of points in a topological space. It allows for the indexing of points with a directed set, enabling a broader approach to convergence, particularly in spaces where sequences may not capture all limit points. Nets facilitate understanding of continuity and the behavior of functions across more complex structures than just metric spaces.
Pasting Lemma: The pasting lemma is a fundamental result in topology that states that if two continuous functions agree on a shared subset, then there exists a continuous function that combines these functions over their respective domains. This concept is crucial for understanding how spaces can be constructed from simpler pieces and ensures that the properties of continuity are preserved during such constructions.
Piecewise continuous: A function is called piecewise continuous if it is continuous on each piece of its domain, except at a finite number of points where it may have discontinuities. In simple terms, a piecewise continuous function can be broken down into sections where it behaves nicely and has well-defined limits, allowing for a more manageable analysis of its properties in the context of continuous functions.
Product topology: Product topology is a way to define a topology on the Cartesian product of two or more topological spaces. In this topology, the basis is formed by products of open sets from each space, which means that a set is open if it can be expressed as a union of such product sets. This concept is essential for understanding how continuous functions behave when dealing with multiple dimensions and helps illustrate the interactions between different topological spaces.
Projection Maps: Projection maps are specific types of continuous functions that take a product space and map it onto one of its factor spaces. They play a critical role in topology, particularly in understanding the relationships between different spaces. Projection maps help illustrate how certain structures can be simplified or analyzed by reducing dimensions, making them essential for studying continuity and properties of functions in the context of topology.
Quasi-continuity: Quasi-continuity refers to a property of functions where the function is continuous at every point of its domain except possibly at a set of points that has measure zero. This means that, unlike standard continuity, it allows for the presence of discontinuities, but these are limited in a specific way. Understanding quasi-continuity helps to bridge gaps between traditional notions of continuity and more generalized concepts, particularly in spaces that exhibit irregular behavior.
Quotient Maps: Quotient maps are continuous functions that relate topological spaces in a way that identifies points in a specific manner, creating a new space from an existing one by collapsing certain subsets into single points. This concept is essential in understanding how spaces can be simplified or altered while maintaining certain properties, thereby connecting the idea of continuity with the formation of new topological structures.
Separation properties: Separation properties are criteria used to distinguish between points and sets in topological spaces, indicating how separated or distinct they are from one another. These properties play a crucial role in understanding the structure of spaces and their continuous functions, as they help determine how functions can behave when mapping between different spaces.
Tietze Extension Theorem: The Tietze Extension Theorem states that if you have a normal topological space and a closed subset within it, any continuous function defined on that closed subset can be extended to a continuous function defined on the entire space. This theorem is significant because it highlights a powerful property of normal spaces, ensuring that continuous functions can be 'lifted' without losing continuity, which is a key aspect when dealing with continuous functions and their properties.
Topological Space: A topological space is a set of points along with a collection of open sets that satisfy certain properties, which help define the concepts of continuity, convergence, and neighborhood in mathematics. This structure allows for the exploration of spaces that may be very different from traditional Euclidean spaces, emphasizing the properties that remain unchanged under continuous transformations.
Uniform continuity: Uniform continuity is a stronger form of continuity that requires a function to maintain a consistent level of closeness between input and output values across its entire domain. Unlike regular continuity, where the level of closeness can vary at different points, uniform continuity guarantees that for any given distance between outputs, there exists a uniform distance between inputs that works everywhere in the domain. This concept connects deeply with how functions behave over their entire range, ensuring that small changes in input result in small changes in output uniformly.