The Central Idea
A topological space \(X\) is compact if every open cover of \(X\) has a finite subcover.
Symbolically, if
\[ X\subseteq\bigcup_{\alpha\in A}U_\alpha \]and every \(U_\alpha\) is open, then compactness guarantees that there exist finitely many indices
\[ \alpha_1,\alpha_2,\ldots,\alpha_n \]such that
\[ \boxed{ X\subseteq U_{\alpha_1}\cup U_{\alpha_2}\cup\cdots\cup U_{\alpha_n} }. \]The key idea is finite control over arbitrary open covers.
What Is a Cover?
A collection of sets
\[ \{U_\alpha\}_{\alpha\in A} \]is a cover of \(X\) if every point of \(X\) lies in at least one member of the collection.
Equivalently,
\[ X\subseteq\bigcup_{\alpha\in A}U_\alpha. \]The sets in the cover may overlap.
What Is an Open Cover?
An open cover is a cover in which every covering set is open in the ambient space or, when working intrinsically, open in the topology under consideration.
For example, the intervals
\[ (-1,2), \qquad (1,4), \qquad (3,6) \]form an open cover of
\[ [0,5]. \]What Is a Subcover?
A subcover is obtained by selecting some of the sets from an existing cover while still covering the entire space.
If
\[ \{U_\alpha\}_{\alpha\in A} \]covers \(X\), then a subcollection
\[ \{U_\beta\}_{\beta\in B}, \qquad B\subseteq A, \]is a subcover when
\[ X\subseteq\bigcup_{\beta\in B}U_\beta. \]What Is a Finite Subcover?
A finite subcover uses only finitely many sets from the original cover.
Thus compactness says:
No matter how large or complicated an open cover is, finitely many of its members already cover the compact space.
Why Compactness Is Not the Same as Finite
Every finite topological space is compact.
But compact spaces can contain infinitely many points.
For example,
\[ [0,1] \]contains infinitely many real numbers, yet it is compact in \(\mathbb{R}\).
Why Every Finite Space Is Compact
Suppose
\[ X=\{x_1,x_2,\ldots,x_n\}. \]If an open cover covers \(X\), choose one covering set containing each point \(x_i\).
At most \(n\) sets are needed.
Therefore every finite space has a finite subcover for every open cover.
A First Compact Example
The closed interval
\[ [0,1] \]is compact in the usual topology on \(\mathbb{R}\).
This fact is a special case of the Heine-Borel theorem.
The Heine-Borel Theorem
In Euclidean space,
\[ \mathbb{R}^n, \]a subset is compact if and only if it is both closed and bounded.
\[ \boxed{ K\subseteq\mathbb{R}^n \text{ compact} \iff K\text{ closed and bounded} }. \]This theorem is extremely useful, but it is specific to Euclidean spaces and closely related settings.
Closed and Bounded on the Real Line
In \(\mathbb{R}\), examples of compact sets include
\[ [0,1], \] \[ [-5,12], \]and any finite union of closed bounded intervals.
Each is closed and bounded.
An Open Interval Is Not Compact
Consider
\[ (0,1). \]It is bounded but not closed in \(\mathbb{R}\).
By Heine-Borel,
\[ \boxed{(0,1)\text{ is not compact}}. \]A Closed Unbounded Set Is Not Compact in \(\mathbb{R}\)
Consider
\[ [0,\infty). \]It is closed but unbounded.
Therefore,
\[ \boxed{[0,\infty)\text{ is not compact}}. \]A Bounded Nonclosed Set Is Not Compact in \(\mathbb{R}\)
Consider
\[ [0,1). \]It is bounded but not closed.
Therefore,
\[ \boxed{[0,1)\text{ is not compact}}. \]An Explicit Open Cover of \((0,1)\)
To see noncompactness directly, consider the open sets
\[ U_n=\left(\frac1n,1\right), \qquad n=2,3,4,\ldots \]Their union covers
\[ (0,1). \]Indeed, every \(x>0\) satisfies
\[ \frac1nWhy That Cover Has No Finite Subcover
Suppose we choose finitely many of the sets
\[ U_{n_1},U_{n_2},\ldots,U_{n_k}. \]Let
\[ N=\max\{n_1,n_2,\ldots,n_k\}. \]Then the union of those finitely many sets is just
\[ \left(\frac1N,1\right). \]Points in
\[ \left(0,\frac1N\right] \]remain uncovered.
Therefore no finite subcover exists.
An Explicit Open Cover of an Unbounded Set
Consider
\[ X=\mathbb{R}. \]The open intervals
\[ U_n=(-n,n), \qquad n=1,2,3,\ldots \]cover \(\mathbb{R}\).
But any finite selection has a largest radius \(N\), so its union is only
\[ (-N,N), \]which does not cover all of \(\mathbb{R}\).
Thus,
\[ \boxed{\mathbb{R}\text{ is not compact}}. \]Compactness Is a Global Property
Openness is tested locally around individual points.
Compactness concerns an entire space and every possible open cover of it.
This makes compactness a fundamentally global property.
Compactness Is Topological
Compactness is preserved by homeomorphisms.
If
\[ X\cong Y, \]then
\[ X\text{ is compact} \iff Y\text{ is compact}. \]Therefore compactness depends on topology, not on a particular coordinate representation.
Continuous Images of Compact Spaces Are Compact
Suppose
\[ f:X\to Y \]is continuous and \(X\) is compact.
Then
\[ \boxed{f(X)\text{ is compact}}. \]This is one of the most useful compactness theorems.
Why Continuous Images Preserve Compactness
Suppose
\[ \{V_\alpha\} \]is an open cover of \(f(X)\).
By continuity, the preimages
\[ f^{-1}(V_\alpha) \]form an open cover of \(X\).
Compactness of \(X\) gives finitely many preimages covering \(X\).
The corresponding finitely many \(V_\alpha\) then cover \(f(X)\).
Closed Subsets of Compact Spaces Are Compact
Let \(X\) be compact and let
\[ F\subseteq X \]be closed.
Then
\[ \boxed{F\text{ is compact}}. \]This theorem works in every topological space.
Why Closed Subsets Inherit Compactness
Suppose an open cover covers \(F\).
Since \(F\) is closed,
\[ X\setminus F \]is open.
Add \(X\setminus F\) to the cover. The enlarged collection covers all of \(X\).
Compactness of \(X\) produces a finite subcover.
Discard \(X\setminus F\) if it appears. The remaining finitely many original sets cover \(F\).
Compact Subsets of Hausdorff Spaces Are Closed
In a Hausdorff space, every compact subset is closed.
Thus,
\[ \boxed{ K\text{ compact in a Hausdorff space} \Longrightarrow K\text{ closed} }. \]Euclidean spaces are Hausdorff, so every compact subset of \(\mathbb{R}^n\) is closed.
What Does Hausdorff Mean?
A topological space is Hausdorff if any two distinct points can be placed inside disjoint open neighborhoods.
If
\[ x\ne y, \]then there exist open sets \(U\) and \(V\) such that
\[ x\in U, \qquad y\in V, \]and
\[ U\cap V=\varnothing. \]This separation property supports many familiar compactness results.
Finite Unions of Compact Sets
If
\[ K_1,K_2,\ldots,K_n \]are compact, then
\[ \boxed{ K_1\cup K_2\cup\cdots\cup K_n \text{ is compact} }. \]A finite number of compact pieces still admits finite control.
Infinite Unions Need Not Be Compact
Each finite interval
\[ [-n,n] \]is compact.
But
\[ \bigcup_{n=1}^{\infty}[-n,n] = \mathbb{R}, \]and \(\mathbb{R}\) is not compact.
Therefore arbitrary unions of compact sets need not be compact.
Arbitrary Intersections of Compact Sets
In a Hausdorff space, compact sets are closed.
Therefore an arbitrary intersection of compact subsets is closed.
If all of those sets lie inside one compact set \(K\), then their intersection is a closed subset of \(K\) and is therefore compact.
The containment condition provides the compact ambient control.
Nested Compact Sets
Suppose
\[ K_1\supseteq K_2\supseteq K_3\supseteq\cdots \]is a nested sequence of nonempty compact subsets of a Hausdorff space.
Under the usual compact-space setting,
\[ \boxed{ \bigcap_{n=1}^{\infty}K_n\ne\varnothing }. \]This is often called the finite intersection principle or nested compact set property.
The Finite Intersection Property
A family of sets has the finite intersection property if every finite subcollection has nonempty intersection.
In a compact space \(X\), if
\[ \{F_\alpha\} \]is a family of closed subsets of \(X\) with the finite intersection property, then
\[ \boxed{ \bigcap_\alpha F_\alpha\ne\varnothing }. \]Compactness and Infinite Constraints
The finite intersection property gives another view of compactness.
Open-cover compactness says:
an infinite covering problem can be reduced to finitely many pieces.
The closed-set version says:
if every finite collection of compatible closed constraints can be satisfied, then all of them can be satisfied simultaneously.
Compactness and Bounded Continuous Functions
Suppose
\[ f:K\to\mathbb{R} \]is continuous and \(K\) is compact.
Since continuous images of compact sets are compact,
\[ f(K) \]is compact in \(\mathbb{R}\).
By Heine-Borel, \(f(K)\) is bounded.
Therefore every continuous real-valued function on a compact space has bounded image.
The Extreme Value Theorem
If
\[ f:K\to\mathbb{R} \]is continuous and \(K\) is compact, then \(f\) attains both a maximum and a minimum.
There exist points
\[ x_{\min},x_{\max}\in K \]such that
\[ f(x_{\min})\le f(x)\le f(x_{\max}) \]for every \(x\in K\).
Why Compactness Matters for Extrema
A continuous function may remain bounded on a noncompact set without attaining its bounds.
For example,
\[ f(x)=x \]on
\[ (0,1) \]has values approaching \(0\) and \(1\), but neither endpoint belongs to the domain.
The function has no minimum and no maximum on \((0,1)\).
Compact Domain Restores Attainment
On the compact interval
\[ [0,1], \]the same function
\[ f(x)=x \]attains
\[ \min f=0 \]and
\[ \max f=1. \]Compactness prevents the extrema from escaping through a missing boundary or toward infinity.
Compactness and Uniform Continuity
A continuous function on a compact metric space is uniformly continuous.
Thus, if
\[ f:K\to Y \]is continuous and \(K\) is compact, then local continuity can be upgraded to a global form of control.
This theorem is often called the Heine-Cantor theorem.
Compactness and Sequences
In metric spaces, compactness has an equivalent sequential description.
A metric space \(X\) is compact if and only if every sequence in \(X\) has a convergent subsequence whose limit belongs to \(X\).
\[ \boxed{ \text{compact} \iff \text{sequentially compact} } \]in metric spaces.
What Is a Subsequence?
Given a sequence
\[ x_1,x_2,x_3,\ldots, \]a subsequence is obtained by selecting terms in their original order:
\[ x_{n_1},x_{n_2},x_{n_3},\ldots \]where
\[ n_1Example: Compactness of \([0,1]\) Through Sequences
Take any sequence
\[ x_1,x_2,x_3,\ldots \]in \([0,1]\).
Because the sequence is bounded, the Bolzano-Weierstrass theorem guarantees a convergent subsequence.
Because \([0,1]\) is closed, the limit remains in \([0,1]\).
This is another manifestation of compactness.
Why \((0,1)\) Fails Sequential Compactness
Consider the sequence
\[ x_n=\frac1n. \]Every term lies in \((0,1)\).
But
\[ x_n\to0, \]and
\[ 0\notin(0,1). \]Every subsequence also tends to \(0\), so no subsequence converges to a point of \((0,1)\).
Why \(\mathbb{R}\) Fails Sequential Compactness
Consider
\[ x_n=n. \]This sequence has no convergent subsequence in \(\mathbb{R}\).
It escapes toward infinity.
Thus noncompactness can arise from missing limit points or from unbounded escape.
Closed and Bounded Capture Two Failure Modes in Euclidean Space
Heine-Borel can be understood through two common ways compactness fails.
- If a set is not closed, sequences may converge to missing boundary points.
- If a set is unbounded, sequences may escape arbitrarily far away.
Closedness prevents missing limits.
Boundedness prevents escape to infinity.
Compactness and Completeness Are Different
A metric space is complete if every Cauchy sequence converges to a point of the space.
Compactness is stronger than completeness in many familiar metric settings.
For example,
\[ \mathbb{R} \]is complete but not compact.
Therefore completeness and compactness are not the same property.
Compact Metric Spaces Are Complete
Every compact metric space is complete.
\[ \boxed{ \text{compact metric space} \Longrightarrow \text{complete} }. \]The converse is false without additional conditions.
Compact Metric Spaces Are Totally Bounded
A metric space is totally bounded if for every \(\varepsilon>0\), finitely many balls of radius \(\varepsilon\) cover the space.
Every compact metric space is totally bounded.
In fact, for metric spaces,
\[ \boxed{ \text{compact} \iff \text{complete and totally bounded} }. \]This is an advanced preview of another useful characterization.
Compactness Is Not Just Boundedness
In general topological spaces, boundedness may not even be defined.
Compactness still makes sense because it uses open sets rather than numerical distance.
Therefore compactness is more fundamental topologically than boundedness.
Compactness Is Not Just Closedness
A set may be closed but fail to be compact.
For example,
\[ \mathbb{R} \]is closed in itself and in the usual ambient setting, but it is not compact.
Closedness alone does not provide finite-cover control.
Compactness Is Relative to the Topology, Not the Ambient Metric Alone
Whether a space is compact depends on its topology.
A homeomorphic copy of a compact space remains compact even if its geometric shape changes dramatically.
This is why compactness belongs naturally to topology.
Subspaces and Compactness
If \(K\subseteq X\), then compactness of \(K\) is determined using the subspace topology on \(K\).
An open cover of \(K\) may be formed from sets open relative to \(K\).
Equivalently, each such set can be written
\[ K\cap U \]for some open \(U\subseteq X\).
Compactness of the Circle
The unit circle
\[ S^1= \{(x,y)\in\mathbb{R}^2:x^2+y^2=1\} \]is closed and bounded in \(\mathbb{R}^2\).
Therefore, by Heine-Borel,
\[ \boxed{S^1\text{ is compact}}. \]Compactness of the Sphere
The unit sphere
\[ S^2= \{(x,y,z)\in\mathbb{R}^3:x^2+y^2+z^2=1\} \]is closed and bounded.
Hence,
\[ \boxed{S^2\text{ is compact}}. \]Compactness of a Closed Disk
Consider
\[ D= \{(x,y):x^2+y^2\le1\}. \]The set is closed and bounded in \(\mathbb{R}^2\).
Therefore,
\[ \boxed{D\text{ is compact}}. \]An Open Disk Is Not Compact
The open unit disk
\[ U= \{(x,y):x^2+y^2<1\} \]is bounded but not closed in \(\mathbb{R}^2\).
Therefore,
\[ \boxed{U\text{ is not compact}}. \]Products of Compact Spaces
The product of two compact spaces is compact.
Thus, if \(X\) and \(Y\) are compact, then
\[ \boxed{X\times Y\text{ is compact}}. \]Finite products follow by repeating the argument.
Example: A Rectangle as a Product
The intervals
\[ [a,b] \]and
\[ [c,d] \]are compact.
Therefore,
\[ [a,b]\times[c,d] \]is compact.
This is the rectangular version of Heine-Borel.
The General Product Theorem
A major theorem called Tychonoff's theorem states that an arbitrary product of compact spaces is compact in the product topology.
This result is far more advanced than the finite-product case.
For now, the important idea is that compactness behaves very well under products.
Compactness and Connectedness Are Independent
A space can be compact without being connected.
For example,
\[ [0,1]\cup[2,3] \]is closed and bounded, so it is compact.
But it is disconnected.
A Space Can Be Connected but Not Compact
The real line
\[ \mathbb{R} \]is connected.
But it is not compact.
Therefore neither connectedness nor compactness implies the other.
Compact and Connected
Many important spaces have both properties.
Examples include:
- closed intervals;
- circles;
- spheres;
- closed disks;
- tori.
Compactness and connectedness together provide strong global structure.
Compactness and the Torus
A standard torus embedded in \(\mathbb{R}^3\) is closed and bounded.
Therefore it is compact.
It is also connected and path connected.
Topological invariants such as compactness, connectedness, and genus describe different aspects of the same surface.
Common Error: Thinking Compact Means Small
Compactness is not a statement about physical size.
The interval
\[ [-10^{100},10^{100}] \]is compact despite being numerically enormous.
Compactness concerns finite subcovers, not everyday notions of compact dimensions.
Common Error: Thinking Compact Means Finite
Compact spaces may contain infinitely or even uncountably many points.
The interval
\[ [0,1] \]is compact and uncountable.
Common Error: Using Closed and Bounded in Every Topological Space
The equivalence
\[ \text{compact} \iff \text{closed and bounded} \]is the Heine-Borel theorem for Euclidean spaces such as \(\mathbb{R}^n\).
It is not the definition of compactness and is not valid in arbitrary topological spaces.
Common Error: Finding One Finite Cover
To prove compactness, it is not enough to exhibit one finite open cover.
Compactness requires that every open cover admit a finite subcover.
The universal quantifier is essential.
Common Error: Finding One Infinite Cover
A compact space may certainly have infinite open covers.
What compactness guarantees is that some finite subcollection of each open cover still covers the space.
Therefore the existence of an infinite cover does not show noncompactness.
Common Error: Showing a Cover Has No Small Subcover
To prove noncompactness using an open cover, one must show that the cover has no finite subcover at all.
Showing that no two or three sets suffice is not enough if four, five, or some other finite number might work.
Common Error: Confusing Compactness with Completeness
The real line is complete but not compact.
Thus,
\[ \text{complete} \centernot\Longrightarrow \text{compact}. \]Compactness imposes stronger global control.
Common Error: Forgetting the Ambient Setting
In Euclidean space, Heine-Borel gives a very convenient test.
In a general topological space, one must return to the open-cover definition or another theorem appropriate to that space.
Always identify the setting before choosing a compactness test.
A Reliable Procedure in \(\mathbb{R}^n\)
- Determine whether the set is closed.
- Determine whether the set is bounded.
- If both hold, invoke Heine-Borel to conclude compactness.
- If either fails, conclude the set is not compact.
- If useful, construct an explicit open cover or sequence explaining the failure.
A Reliable General Topological Procedure
- Start with an arbitrary open cover.
- Use the structure of the space to reduce the cover.
- Show that finitely many members suffice.
- Alternatively, apply a known compactness theorem.
- To prove noncompactness, construct one open cover having no finite subcover.
Example: Classify \([-3,7]\)
The interval is closed.
It is bounded.
Therefore, by Heine-Borel,
\[ \boxed{[-3,7]\text{ is compact}}. \]Example: Classify \((-3,7]\)
The set is bounded but not closed in \(\mathbb{R}\).
Therefore,
\[ \boxed{(-3,7]\text{ is not compact}}. \]Example: Classify \(\mathbb{Z}\)
The integers are closed in \(\mathbb{R}\).
But they are unbounded.
Therefore,
\[ \boxed{\mathbb{Z}\text{ is not compact in }\mathbb{R}}. \]Example: Classify a Finite Set
Let
\[ F=\{-4,0,3,11\}. \]Every finite topological space is compact.
Also, as a subset of \(\mathbb{R}\), \(F\) is closed and bounded.
Therefore,
\[ \boxed{F\text{ is compact}}. \]Example: Compact Image
Let
\[ f:[0,2]\to\mathbb{R} \]be defined by
\[ f(x)=x^2. \]The domain \([0,2]\) is compact and \(f\) is continuous.
Therefore,
\[ f([0,2])=[0,4] \]is compact.
Example: Compactness and Extrema
Consider
\[ f(x)=x^2-4x+7 \]on
\[ [0,5]. \]The domain is compact and the function is continuous.
Therefore \(f\) must attain both a minimum and a maximum somewhere on the interval.
Compactness guarantees existence even before we calculate their locations.
Practice 1: Closed Interval
Is
\[ [-2,9] \]compact in \(\mathbb{R}\)?
Practice 2: Open Interval
Is
\[ (-2,9) \]compact in \(\mathbb{R}\)?
Practice 3: Half-Open Interval
Is
\[ [-2,9) \]compact in \(\mathbb{R}\)?
Practice 4: Unbounded Closed Set
Is
\[ [3,\infty) \]compact in \(\mathbb{R}\)?
Practice 5: Circle
Is
\[ S^1= \{(x,y):x^2+y^2=1\} \]compact in \(\mathbb{R}^2\)?
Practice 6: Finite Set
Is
\[ \{1,4,8,15\} \]compact?
Practice 7: Continuous Image
Suppose \(X\) is compact and
\[ f:X\to Y \]is continuous.
What can be concluded about \(f(X)\)?
Practice 8: Closed Subset
Suppose \(X\) is compact and \(F\subseteq X\) is closed.
What can be concluded about \(F\)?
Practice 9: Sequence Test
Consider the sequence
\[ x_n=n \]in \(\mathbb{R}\).
Explain how it helps show that \(\mathbb{R}\) is not compact.
Practice 10: Heine-Borel
Complete:
\[ K\subseteq\mathbb{R}^n \text{ compact} \iff \boxed{\phantom{\text{closed and bounded}}}. \]Practice Answers
Practice 1
The interval is closed and bounded.
\[ \boxed{[-2,9]\text{ is compact}}. \]Practice 2
The interval is bounded but not closed.
\[ \boxed{(-2,9)\text{ is not compact}}. \]Practice 3
The interval is bounded but not closed.
\[ \boxed{[-2,9)\text{ is not compact}}. \]Practice 4
The set is closed but unbounded.
\[ \boxed{[3,\infty)\text{ is not compact}}. \]Practice 5
The unit circle is closed and bounded in \(\mathbb{R}^2\).
\[ \boxed{S^1\text{ is compact}}. \]Practice 6
Every finite topological space is compact.
\[ \boxed{\{1,4,8,15\}\text{ is compact}}. \]Practice 7
Continuous images of compact spaces are compact.
\[ \boxed{f(X)\text{ is compact}}. \]Practice 8
Closed subsets of compact spaces are compact.
\[ \boxed{F\text{ is compact}}. \]Practice 9
The sequence
\[ 1,2,3,\ldots \]has no convergent subsequence in \(\mathbb{R}\).
Since compact metric spaces require every sequence to have a convergent subsequence with limit in the space, \(\mathbb{R}\) is not compact.
Practice 10
\[ \boxed{\text{closed and bounded}}. \]What to Remember
- A space is compact when every open cover has a finite subcover.
- Every finite space is compact.
- In \(\mathbb{R}^n\), compact is equivalent to closed and bounded.
- Continuous images of compact spaces are compact.
- Closed subsets of compact spaces are compact.
- Compact subsets of Hausdorff spaces are closed.
- Finite unions of compact sets are compact.
- Infinite unions of compact sets need not be compact.
- In metric spaces, compactness is equivalent to sequential compactness.
- Continuous real-valued functions on compact spaces are bounded and attain their extrema.
- Compactness and connectedness are independent properties.
- Compactness is topological; boundedness is not its general definition.
The essential compactness principle is:
\[ \boxed{ \text{arbitrary open cover} \longrightarrow \text{finite subcover} }. \]Compactness allows topology to extract finite structure from potentially infinite data.
Where We Go Next
Compactness joins connectedness as one of topology's major global properties. Connectedness asks whether a space can be separated; compactness asks whether its open-cover behavior can always be reduced to finitely many pieces.
These ideas lead naturally toward continuity, homeomorphisms, local and global structure, compact manifolds, convergence, and qualitative classification. Compactness also becomes a central tool throughout analysis, where it guarantees boundedness, extrema, convergent subsequences, and other forms of global control.