In addition, cantor sets are uncountable, may have 0 or positive lebesgue measures, and are nowhere dense. Later, we will see that the cantor set has many other interesting properties. The abstract concepts of metric spaces are often perceived as difficult. With applications features a thorough introduction to cantor sets and applies these sets as a bridge between real analysis, probability, topology, and algebra. A pair, where is a metric on is called a metric space. Pdf an alternate characterization of the cantor set. Also any subset of a metric space is a metric space. Named for the mathemati cian georg cantor, this set is a fractal a type of selfsimilar object and possesses many strange properties. Rather than passing quickly from the definition of a metric to the more abstract concepts of convergence and continuity, the author takes the concrete notion of. A cantor set is a perfect 0dimensional compact metric space. Which category a set has depends on the space within which it is considered.
The baire category theorem let x be a metric space. Some properties of the cantor set unt digital library. This space is nonempty, compact, metric, perfect, and has a countable base for the topol ogy consisting of clopen. Defn a subset c of a metric space x is called closed if its complement is open in x. Quasisymmetric invariant properties of cantor metric spaces. The cantor set is uncountable trinity college dublin.
An open in cantor space is a collection g g of finite sequences of binary digits that is a subset of the free monoid \mathbbb such that. A metric space is said to be locally compact if every point has a compact neighborhood. Pdf continuous images of cantors ternary set researchgate. Let c be the projection of the standard ternary cantor set on the unit interval to the circle. A metric space is just a set x equipped with a function d of two variables which measures the distance between points. Let cx denote the space of all realvalued, bounded continuous functions on a topological space x. Cantors theorem basic idea cantors theorem explained proof. If card x c, then x contains a homeomorphic image of the cantor set l. The two standard proofs of this are given in the references below. We have seen above that the cantor set is a totally disconnected perfect compact metric space. Euclidean spaces are locally compact, but infinitedimensional banach.
A metric space m is totally disconnected if, for any 0 and. A totally disconnected space is a space x where for every two points x1. Notes on metric spaces these notes are an alternative to the textbook, from and including closed sets and open sets page 58 to and excluding cantor sets page 95 1 the topology of metric spaces assume m is a metric space with distance function d. Every compact metric space is second countable, and is a continuous image of the cantor set.
Xthe number dx,y gives us the distance between them. Pdf metric spaces notes free download tutorialsduniya. Cantor set and its properties university of california. A counterexample to this claim is the cantor set c. For a metric space let us consider the space of all nonempty closed bounded subset of with the following metric. If s is the cring generated by the compact sets of the compact metric space x, and if p. A systematic and integrated approach to cantor sets and their applications to various branches of mathematics the elements of cantor sets. An alternate characterization of the cantor set article pdf available in proceedings of the american mathematical society 531. Except in some degenerate situations the space xis in nite, indeed uncountable, and homeomorphic to the cantor set. X a cantor set if c is homeomorphic to the cantor space 2n. This will take us through several mathematical disciplines. The cantor set, also known as the cantor nomiddlethird set, can be constructed as follows.
Totally disconnected means the connected components the maximal connected subsets of the space are all points. Anatole katok, jeanpaul thouvenot, in handbook of dynamical systems, 2006. Complete metric space of compact sets with hausdor distance hutchnsons theorem on attractors of contracting ifs examples. The proof of cantor s intersection theorem on nested compact sets. The cantor set and the cantor function tma4225 foundations of analysis clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. U nofthem, the cartesian product of u with itself n times.
In this course, the objective is to develop the usual idea of distance into an abstract form on any set of objects, maintaining its inherent characteristics. Xis a compact metric space when we agree that two paths are close if they agree on a long initial segment. A metric space is a set equipped with a distance function, which provides a measure of distance between any two points in the set. August 2, 2016 this assignment is about the cantor set, a remarkable subset of 0. One represents a metric space s s s with metric d d d as the pair s, d s, d s, d. The author fills a gap in the current literature by providing an introductory and integrated. See cantor space for more on spaces homeomorphic to the cantor set. Local trivializations of suspended minimal cantor systems.
It also includes an application of these cantor subspaces to a characterization of the countability of closed subsets of r in terms of some simple exterior measures. Consider a as the classical cantor set in the unit interval 0, 1, and extend 4 to a continuous function defined on the whole interval by linear interpolation. Cantors set is the set c left after this procedure of deleting the. Last month, i wrote about the cantor set, a mathematical space that is an interesting mix of small and large. Named for the mathematician georg cantor, this set is a fractal a type of selfsimilar object and possesses many strange. The set of all rational numbers q is not a nowhere dense set. Can every compact metric space be realized as the continuous image of a cantor set. The cantor ternary set is created by repeatedly deleting the.
Several interesting remarks and theorems relating to cantor sets are then demonstrated, and their connection to real analysis and general topology explored. If a, b is one of these 832 surjeciivity of the cantor set november. A 1910 theorem of brouwer characterizes the cantor set as the unique totally disconnected, compact metric space without isolated points. A nonempty hausdorff topological space is compact metrizable if and only if it is a continuous image of a cantor space. A point in cantor space is an infinite sequence of binary digits. Theorem, there is a continuous function 4 from the cantor set a onto the compact metric space 0, 1d. Topologycantor space wikibooks, open books for an open. Suppose that x is a complete metric space, and is a sequence of nonempty closed nested subsets of x whose diameters tend to zero. The cantor set and the cantor function tma4225 foundations of analysis. The cantor ternary set is created by repeatedly deleting the open middle thirds of a set of line segments. In this course, the objective is to develop the usual idea of distance into an abstract form on any set of objects, maintaining its inherent characteristics, and the resulting consequences. Often, if the metric dis clear from context, we will simply denote the metric space x.
Applications of the universal surjectivity of the cantor set. Suppose we x a linear order on the set of edges into each vertex. An open in cantor space is a collection g g of finite sequences of binary digits that is a subset of the free monoid. Local trivializations of suspended minimal cantor systems and the stable orbitbreaking subalgebra j. Pdf the cantor set is an interesting example of an uncountable set of. The particular distance function must satisfy the following conditions. The latter result is due to pavel alexandrov and urysohn. A metric space is a set x where we have a notion of distance. The answer is yes, and the theory is called the theory of metric spaces.
The cantor set is the intersection of this decreasing or nested sequence of sets and so is also closed. The distance function, known as a metric, must satisfy a collection of axioms. Open and closed sets defn if 0, then an open neighborhood of x is defined to be the set b x. Pdf the hausdorffalexandroff theorem states that any compact metric space is the. The cantor intersection theorem for complete metric spaces. A metric space x is said to be connected if it contains no two disjoint nonempty open sets. Cantor set and its properties zhixing guo university of california, santa barbara april 23, 2014. A metric space is a cantor space if it is compact, perfect and totally disconnected.
Named for the mathemati cian georg cantor, this set is a fractal a type of self similar object and possesses many strange properties. A metric space is a set xtogether with a metric don it, and we will use the notation x. Recall that a set x is countable if there is a sequence of. The cantor set is an interesting example of an uncountable set of measure zero and has many interesting properties and consequences in the fields of set theory, topology, and fractal theory. Metrisable means there exists a metric for this space. It is well known that if c is the cantor middle thirds set and x a nonempty compact metric space, there is a continuous surjection. Moreover, from the remarks above it follows that each x 2c has one and only one ternary expansion using only 0s and 2s. Pdf on sep 30, 2014, lakshmi kanta dey and others published cantors intersection theorem and baires category theorem in generalized metric space find, read and cite all the research you need. In a complete metric space, the following variant of cantor s intersection theorem holds. This book offers a unique approach to the subject which gives readers the advantage of a new perspective on ideas familiar from the analysis of a real line. Twotopologicaluniquenesstheoremsfor spacesofrealnumbers. Every open subset of a complete and perfect space contains a cantor space. Thus the theorem states that the combination of these conditions leaves a space homeomorphic to the cantor set. Set theory from cantor to cohen akihiro kanamori set theory is an autonomous and sophisticated.
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