Compact sets are a fundamental concept in mathematics, particularly in the field of topology. A compact set is a set that has certain properties that make it "small" or "bounded" in some sense. In this article, we will explore the 5 key properties of a compact set in math, explaining each property in detail and providing examples to illustrate their significance.
Key Points
- A compact set is closed and bounded.
- Compact sets have the finite intersection property.
- Compact sets are sequentially compact.
- Compact sets have the Bolzano-Weierstrass property.
- Compact sets are preserved under continuous functions.
Property 1: Closed and Bounded
A compact set is a set that is both closed and bounded. A set is closed if it contains all its limit points, meaning that if a sequence of points in the set converges to a point, then that point is also in the set. A set is bounded if it is contained in a ball of finite radius, meaning that there exists a point and a radius such that every point in the set is within that radius of the point.
For example, the set of points in $\mathbb{R}^2$ that satisfy $x^2 + y^2 \leq 1$ is a closed and bounded set, and therefore compact. On the other hand, the set of points in $\mathbb{R}^2$ that satisfy $x^2 + y^2 < 1$ is bounded but not closed, and therefore not compact.
Heine-Borel Theorem
The Heine-Borel theorem states that a set in $\mathbb{R}^n$ is compact if and only if it is closed and bounded. This theorem provides a useful characterization of compact sets in finite-dimensional spaces.
| Set | Closed | Bounded | Compact |
|---|---|---|---|
| $\{x \in \mathbb{R}^2 : x^2 + y^2 \leq 1\}$ | Yes | Yes | Yes |
| $\{x \in \mathbb{R}^2 : x^2 + y^2 < 1\}$ | No | Yes | No |
Property 2: Finite Intersection Property
A collection of sets has the finite intersection property if the intersection of any finite subcollection of sets is non-empty. Compact sets have the finite intersection property, meaning that if we have a collection of closed sets in a compact set, then the intersection of any finite subcollection of sets is non-empty.
For example, consider the collection of sets $\{\{1 - \frac{1}{n} : n \in \mathbb{N}\}\}$. This collection has the finite intersection property, but the intersection of all sets in the collection is empty.
Example of Finite Intersection Property
Let $X$ be a compact set and let $\{F_\alpha\}$ be a collection of closed sets in $X$. Suppose that $\bigcap_{\alpha \in A} F_\alpha = \emptyset$ for some subset $A$ of the index set. Then there exists a finite subset $A_0$ of $A$ such that $\bigcap_{\alpha \in A_0} F_\alpha = \emptyset$.
Property 3: Sequential Compactness
A set is sequentially compact if every sequence of points in the set has a convergent subsequence. Compact sets are sequentially compact, meaning that if we have a sequence of points in a compact set, then there exists a subsequence that converges to a point in the set.
For example, consider the set of points in $\mathbb{R}$ that satisfy $0 \leq x \leq 1$. This set is compact and therefore sequentially compact. The sequence $\{1 - \frac{1}{n}\}$ is a sequence of points in this set, and it has a convergent subsequence that converges to $1$.
Proof of Sequential Compactness
Let $X$ be a compact set and let $\{x_n\}$ be a sequence of points in $X$. Let $U$ be an open cover of $X$. Then there exists a finite subcover $\{U_1, \ldots, U_k\}$ of $U$. For each $i$, let $n_i$ be the smallest integer such that $x_n \in U_i$ for all $n \geq n_i$. Then there exists $i$ such that $x_n \in U_i$ for infinitely many $n$. Let $\{x_{n_j}\}$ be a subsequence of $\{x_n\}$ such that $x_{n_j} \in U_i$ for all $j$. Then $\{x_{n_j}\}$ is a convergent subsequence.
Property 4: Bolzano-Weierstrass Property
A set has the Bolzano-Weierstrass property if every infinite subset of the set has a limit point in the set. Compact sets have the Bolzano-Weierstrass property, meaning that if we have an infinite subset of a compact set, then there exists a limit point of the subset that is also in the set.
For example, consider the set of points in $\mathbb{R}$ that satisfy $0 \leq x \leq 1$. This set is compact and therefore has the Bolzano-Weierstrass property. The set of points $\{1 - \frac{1}{n} : n \in \mathbb{N}\}$ is an infinite subset of this set, and it has a limit point $1$ that is also in the set.
Example of Bolzano-Weierstrass Property
Let $X$ be a compact set and let $A$ be an infinite subset of $X$. Suppose that $A$ has no limit points in $X$. Then $A$ is a discrete subspace of $X$, and therefore $A$ is closed in $X$. But then $X \setminus A$ is an open set that contains no points of $A$, contradicting the fact that $A$ is infinite.
Property 5: Preservation under Continuous Functions
Compact sets are preserved under continuous functions, meaning that if we have a continuous function from a compact set to another space, then the image of the compact set is also compact.
For example, consider the function $f(x) = x^2$ from the set of points in $\mathbb{R}$ that satisfy $0 \leq x \leq 1$ to $\mathbb{R}$. This function is continuous, and the image of the set is the set of points in $\mathbb{R}$ that satisfy $0 \leq x \leq 1$, which is compact.
Proof of Preservation
Let $X$ be a compact set and let $f: X \to Y$ be a continuous function. Let $U$ be an open cover of $f(X)$. Then $\{f^{-1}(U) : U \in U\}$ is an open cover of $X$. Let $\{f^{-1}(U_1), \ldots, f^{-1}(U_k)\}$ be a finite subcover. Then $\{U_1, \ldots, U_k\}$ is a finite cover of $f(X)$, and therefore $f(X)$ is compact.
What is a compact set in math?
+A compact set is a set that has certain properties that make it "small" or "bounded" in some sense. It is a set that is both closed and bounded, and it has the finite intersection property, is sequentially compact, has the Bolzano-Weierstrass property, and is preserved under continuous functions.
What is the Heine-Borel theorem?
+The Heine-Borel theorem states that a set in $\mathbb{R}^n$ is compact if and only if it is closed and bounded.
What is the finite intersection property?
+A collection of sets has the finite intersection property if the intersection of any finite subcollection of sets is non-empty.
What is sequential compactness?
+A set is sequentially compact if every sequence of points in the set has a convergent subsequence.
In conclusion, compact sets are a fundamental concept in mathematics, and they have numerous applications in analysis, topology, and other fields. The 5 key properties of a compact set, including being closed and bounded, having the finite intersection property, being sequentially compact, having the Bolzano-Weierstrass property, and being preserved under continuous functions, make them a crucial tool for mathematicians and scientists.
