Problem 8
Question
(The Cantor set.) Let \(C\) denote the set of real numbers \(x\) of the form $$ x=\sum_{n=1}^{\infty} \alpha_{n} 3^{-n}, \quad \text { where } \alpha_{n}=0 \text { or } \alpha_{n}=2 $$ Show that \(C=\bigcap C_{n}\), where \(C_{1}=[0,1]\) and where \(C_{n+1}\) is obtained from \(C_{n}\) by deleting the (open) middle third of each of the intervals that belong to \(C_{n}\). Deduce from this that \(C\) (in the relative topology induced from \(\mathbb{R}\) ) is a compact Hausdorff space. Show that \(C\) as a subset of \(\mathbb{R}\) has empty interior, but no isolated points. Show that every \(x\) in \(C\) has a unique expression \(x=\sum \alpha_{n} 3^{-n}\) and that \(C\) is homeomorphic with the product space \(\\{0,1\\}^{N}\) (and thus uncountable). Show that the map \(f: C \rightarrow[0,1]\) given by \(f(x)=\sum \alpha_{n} 2^{-n-1}\), where \(x=\sum \alpha_{n} 3^{-n}\), is continuous and surjective.
Step-by-Step Solution
Verified Answer
The Cantor set is compact, Hausdorff, has empty interior, no isolated points, and is homeomorphic to \(\{0,1\}^{\mathbb{N}}\) with a continuous, surjective map \( f: C \to [0,1] \).
1Step 1: Understanding the Cantor Set
The Cantor set consists of numbers expressed in base 3 using only the digits 0 and 2. We want to show that \( C = \bigcap C_n \). The set \( C_1 = [0, 1] \) and \( C_{n+1} \) is derived by removing the middle third from each interval in \( C_n \). This iterative process results in a fractal dimension, with the Cantor set consisting of all such real numbers obtained via this construction.
2Step 2: Establishing Compactness of the Cantor Set
A space is compact if it is closed and bounded, and as a subset of \(\mathbb{R}\), the Cantor set is bounded within \([0,1]\). Each \( C_n \) is closed because it is a finite union of closed intervals after removing open intervals. The entire Cantor set \( C \), being an intersection of these closed \( C_n \) sets, is closed. As \( C \) is both closed and bounded, it is compact.
3Step 3: Confirming the Hausdorff Property
In \(\mathbb{R}\), a metric space, every subset is Hausdorff. Thus, any two points in \( C \) can be separated by open neighborhoods. This makes the Cantor set a compact Hausdorff space, as any compact subset of a Hausdorff space is also Hausdorff.
4Step 4: Showing Empty Interior and No Isolated Points
The interior of the Cantor set is empty as it contains no open interval; each attempt to form an interior interval is thwarted by the removal of middle thirds in its construction. However, there are no isolated points, as between any two points, another point in the Cantor set can be found, due to its dense nature in itself.
5Step 5: Uniqueness of Ternary Representation
Every \( x \in C \) has a unique expansion \( x = \sum \alpha_n \frac{2}{3^n} \). This results from choosing digits \( \alpha_n = 0 \text{ or } 2 \), avoiding the non-uniqueness of ternary expansion which occurs with the digit 1. This unique sequence maps each element of \( C \) to a distinct infinite binary sequence, forming a bijection with the product space \( \{0, 1\}^\mathbb{N} \).
6Step 6: Demonstrating Homeomorphism to a Product Space
The Cantor set \( C \) is homeomorphic to \( \{0,1\}^{\mathbb{N}} \) because each element of \( C \) corresponds to a unique infinite sequence of 0s and 1s. This one-to-one and continuous mapping preserves structure, proving \( C \) is uncountable as it matches the cardinality of \( \{0,1\}^{\mathbb{N}} \), a well-known uncountable set.
7Step 7: Proving Continuity and Surjectivity of Map f
Define \( f(x) = \sum \alpha_n 2^{-(n+1)} \). The map \( f \), making each ternary digit \( \alpha_n \) influence the binary fraction contribution, is continuous as small changes in \( x \) (from a ternary perspective) lead to small changes in the binary sum. It is surjective because for any number \( y \in [0,1] \), there is an \( x \) in \( C \) that maps to it by appropriately choosing the sequences \( \alpha_n \).
Key Concepts
Compact Hausdorff SpaceTernary RepresentationHomeomorphic MappingTopologyMetric Space
Compact Hausdorff Space
The concept of a compact Hausdorff space is a fundamental idea in topology, capturing certain valuable properties of a space. In simple terms, a space is **compact** if every open cover of the space has a finite subcover. This means that no matter how you choose to "cover" the entire space with open sets, you can always find a finite number of these sets that still cover the entire space. In the case of the Cantor set, it is compact because it is an intersection of closed and bounded sets; these properties imply compactness in the Euclidean space \( \mathbb{R} \).
An important feature of compact spaces is that they behave similarly to closed and bounded sets from Euclidean space, as guaranteed by the Heine-Borel theorem. The **Hausdorff property** adds another layer by ensuring that points can be separated by neighborhoods. In \( \mathbb{R} \), any subset is Hausdorff, and the Cantor set, as a subset, inherits this property. Thus, the Cantor set is both compact and Hausdorff, making it a compact Hausdorff space.
An important feature of compact spaces is that they behave similarly to closed and bounded sets from Euclidean space, as guaranteed by the Heine-Borel theorem. The **Hausdorff property** adds another layer by ensuring that points can be separated by neighborhoods. In \( \mathbb{R} \), any subset is Hausdorff, and the Cantor set, as a subset, inherits this property. Thus, the Cantor set is both compact and Hausdorff, making it a compact Hausdorff space.
Ternary Representation
The **ternary representation** describes numbers based on powers of three. For a number in this system, digits are typically 0, 1, or 2. Interestingly, the Cantor set uses a special form of ternary representation where the digit 1 is excluded. Each number in the Cantor set is expressed as \( x = \sum_{n=1}^{\infty} \alpha_{n} 3^{-n} \), where \( \alpha_n \) is 0 or 2.
Because the number 1 is never used, each ternary number in the Cantor set corresponds uniquely to an item within it, avoiding any ambiguity from using 1, which might otherwise allow alternate representations. This distinctive ternary form is crucial for ensuring every number in the Cantor set can be matched with its specific position within this compact Hausdorff space.
Because the number 1 is never used, each ternary number in the Cantor set corresponds uniquely to an item within it, avoiding any ambiguity from using 1, which might otherwise allow alternate representations. This distinctive ternary form is crucial for ensuring every number in the Cantor set can be matched with its specific position within this compact Hausdorff space.
Homeomorphic Mapping
When we discuss **homeomorphic mapping**, we're exploring the idea that two spaces are structurally "the same" even if they might appear different at first glance. Two spaces are homeomorphic if there exists a continuous function with a continuous inverse between them. This implies that the spaces have the same "topological" features, such as connectedness or number of holes.
The Cantor set is homeomorphic to the product space \( \{0,1\}^{\mathbb{N}} \. \) This means there's a continuous, one-to-one mapping from the Cantor set numbers (expressed in their unique ternary forms) to infinite binary sequences. This property not only establishes a connection between the Cantor set and a well-known space in set theory but also helps demonstrate its uncountable nature by linking it to an uncountably large set.
The Cantor set is homeomorphic to the product space \( \{0,1\}^{\mathbb{N}} \. \) This means there's a continuous, one-to-one mapping from the Cantor set numbers (expressed in their unique ternary forms) to infinite binary sequences. This property not only establishes a connection between the Cantor set and a well-known space in set theory but also helps demonstrate its uncountable nature by linking it to an uncountably large set.
Topology
**Topology** is the mathematical study of shapes and spaces and their properties that remain unchanged through continuous deformations. In the context of the Cantor set, topology helps us understand its structure through concepts such as compactness, connectedness, and continuity.
The Cantor set's topology is derived from its construction process of repeatedly removing middle thirds from intervals, leaving behind a collection of points defined by their unique ternary representations. This process results in a set with fascinating topological features. It is disconnected yet perfectly compact, having no interior, yet no isolated points, a common scenario seen in the Cantor set.
The Cantor set also provides insights into how spaces can be manipulated and understood through purely topological properties, emphasizing the value of looking at spaces from different angles and understanding their core properties rather than just their geometric configurations.
The Cantor set's topology is derived from its construction process of repeatedly removing middle thirds from intervals, leaving behind a collection of points defined by their unique ternary representations. This process results in a set with fascinating topological features. It is disconnected yet perfectly compact, having no interior, yet no isolated points, a common scenario seen in the Cantor set.
The Cantor set also provides insights into how spaces can be manipulated and understood through purely topological properties, emphasizing the value of looking at spaces from different angles and understanding their core properties rather than just their geometric configurations.
Metric Space
A **metric space** is simply a set where a notion of distance between any two points is defined. It's one fundamental way mathematicians explore and analyze the structure of a space. For the Cantor set, when considered as a subset of \mathbb{R}\, a metric space naturally arises with the standard Euclidean distance used as the metric.
In general, the properties of metric spaces facilitate a deeper understanding of convergence, continuity, and compactness. In the case of the Cantor set, having a metric allows us to reason about how points are spaced within \( \mathbb{R} \) and contributes to our understanding of its structure, including why it retains its compactness.
Thus, while the Cantor set's intriguing properties might be seeable on a higher topological level, the underpinning metric framework provides a clear, methodical path to analyze its peculiar and fascinating characteristics.
In general, the properties of metric spaces facilitate a deeper understanding of convergence, continuity, and compactness. In the case of the Cantor set, having a metric allows us to reason about how points are spaced within \( \mathbb{R} \) and contributes to our understanding of its structure, including why it retains its compactness.
Thus, while the Cantor set's intriguing properties might be seeable on a higher topological level, the underpinning metric framework provides a clear, methodical path to analyze its peculiar and fascinating characteristics.
Other exercises in this chapter
Problem 7
(The Sorgenfrey line.) Give the set \(\mathbb{R}\) the topology \(\tau\) for which a basis consists of the half-open intervals \([y, z[\), where \(y\) and \(z\)
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(The Sorgenfrey plane.) Give the set \(\mathbb{R}^{2}\) the topology \(\tau^{2}\), for which a basis consist of products of half-open intervals \(\left[y_{1}, z
View solution Problem 8
A set is called countable (or countably infinite) if it has the same cardinality (cf. E 1.1.6) as the set No natural numbers. Show that there is a well-ordered
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