The Central Idea
A space is connected when it cannot be split into two nonempty open pieces that do not overlap.
A space is path connected when every pair of points can be joined by a continuous path that remains inside the space.
Path connectedness is stronger:
\[ \boxed{\text{path connected}\Longrightarrow\text{connected}}. \]The reverse implication is not always true.
What Does "One Piece" Mean?
A picture may suggest that a set is one piece, but topology needs a definition that works even when no useful picture exists.
Connectedness captures the idea of "one piece" through open sets.
Path connectedness captures it through continuous motion.
These two ideas agree in many familiar geometric settings, but they are not logically identical.
Separations
Let \(X\) be a topological space.
A separation of \(X\) is a pair of subsets \(U\) and \(V\) such that
\[ X=U\cup V, \] \[ U\cap V=\varnothing, \]and both \(U\) and \(V\) are nonempty and open in \(X\).
If such a pair exists, the space is disconnected.
Connected Spaces
A topological space \(X\) is connected if no separation of \(X\) exists.
Symbolically, there do not exist nonempty open sets \(U,V\subseteq X\) such that
\[ X=U\cup V \]and
\[ U\cap V=\varnothing. \]Therefore,
\[ \boxed{ X\text{ connected} \iff X\text{ has no separation} }. \]Disconnected Spaces
A space is disconnected if it does admit a separation.
That means the space can be decomposed as
\[ X=U\cup V \]where \(U\) and \(V\) are:
- open in \(X\);
- nonempty;
- disjoint.
Example: Two Separated Intervals
Consider
\[ X=(-2,-1)\cup(1,3). \]Let
\[ U=(-2,-1) \]and
\[ V=(1,3). \]Then \(U\) and \(V\) are nonempty, disjoint, open in the subspace \(X\), and satisfy
\[ X=U\cup V. \]Therefore,
\[ \boxed{X\text{ is disconnected}}. \]Why Relative Openness Matters
When testing whether a subset \(X\) is connected, the relevant open sets are open in the topology of \(X\) itself.
They need not be open in a larger ambient space.
Connectedness is therefore always interpreted relative to the topology on the space being studied.
Connectedness and Clopen Sets
A set is clopen when it is both open and closed.
A space \(X\) is connected exactly when its only clopen subsets are
\[ \varnothing \]and
\[ X. \]Thus,
\[ \boxed{ X\text{ connected} \iff \varnothing\text{ and }X\text{ are the only clopen subsets of }X }. \]Why a Nontrivial Clopen Set Disconnects a Space
Suppose \(A\subseteq X\) is nonempty, not equal to \(X\), and clopen.
Because \(A\) is closed, its complement
\[ X\setminus A \]is open.
Since \(A\) itself is open,
\[ X=A\cup(X\setminus A) \]gives a separation.
Therefore \(X\) is disconnected.
Intervals in \(\mathbb{R}\) Are Connected
One of the fundamental theorems of elementary topology is:
\[ \boxed{\text{Every interval in }\mathbb{R}\text{ is connected}}. \]This includes:
- open intervals;
- closed intervals;
- half-open intervals;
- rays;
- the entire real line;
- single-point intervals.
Why Intervals Behave Differently from Gapped Sets
An interval has no missing point between two of its points.
If
\[ aThis "no gaps" property is exactly what characterizes connected subsets of the real line.
Connected Subsets of the Real Line
A subset \(A\subseteq\mathbb{R}\) is connected if and only if it is an interval, including degenerate intervals consisting of a single point.
\[ \boxed{ A\subseteq\mathbb{R}\text{ connected} \iff A\text{ is an interval} }. \]This provides a complete classification of connected subsets of \(\mathbb{R}\).
Example: A Closed Interval
The set
\[ [0,4] \]is an interval.
Therefore,
\[ \boxed{[0,4]\text{ is connected}}. \]Example: A Half-Open Interval
The set
\[ [0,4) \]is also an interval.
Therefore,
\[ \boxed{[0,4)\text{ is connected}}. \]Connectedness is not the same as being open or closed.
Example: Removing a Point Can Disconnect an Interval
Start with
\[ X=(-1,1). \]Remove \(0\):
\[ X\setminus\{0\} = (-1,0)\cup(0,1). \]The result has two disjoint nonempty open pieces.
Therefore,
\[ \boxed{(-1,1)\setminus\{0\}\text{ is disconnected}}. \]A Single Point Is Connected
A one-point space
\[ X=\{p\} \]cannot be split into two nonempty disjoint open subsets.
Therefore every singleton space is connected.
The Empty Space
Under the standard definition, the empty space is connected because no separation into two nonempty open sets exists.
\[ \boxed{\varnothing\text{ is connected}}. \]Some informal discussions focus only on nonempty spaces, but the standard topological definition includes the empty space as connected.
What Is a Path?
A path in a topological space \(X\) from a point \(x\) to a point \(y\) is a continuous function
\[ \gamma:[0,1]\to X \]such that
\[ \gamma(0)=x \]and
\[ \gamma(1)=y. \]The parameter \(t\in[0,1]\) represents progress along the path.
Why the Interval \([0,1]\) Is Used
The interval
\[ [0,1] \]provides a standard domain for describing continuous motion from a starting point to an ending point.
Other intervals could be used, but \([0,1]\) is conventional and convenient.
Path Connected Spaces
A topological space \(X\) is path connected if for every pair of points
\[ x,y\in X, \]there exists a continuous path
\[ \gamma:[0,1]\to X \]satisfying
\[ \gamma(0)=x \]and
\[ \gamma(1)=y. \]Path Connectedness as Continuous Travel
Informally, path connectedness means:
you can travel continuously from any point of the space to any other point without leaving the space.
The path may curve, double back, or take a complicated route.
It only needs to be continuous and remain inside the space.
Example: An Interval Is Path Connected
Let \(I\subseteq\mathbb{R}\) be an interval, and choose
\[ x,y\in I. \]Define
\[ \gamma(t)=(1-t)x+ty. \]Then
\[ \gamma(0)=x \]and
\[ \gamma(1)=y. \]Because every point between \(x\) and \(y\) lies in the interval, \(\gamma(t)\in I\) for all \(t\in[0,1]\).
Therefore every interval is path connected.
Straight-Line Paths in Euclidean Space
In \(\mathbb{R}^n\), the straight-line path from \(x\) to \(y\) is
\[ \boxed{ \gamma(t)=(1-t)x+ty }. \]This formula moves continuously from \(x\) at \(t=0\) to \(y\) at \(t=1\).
Convex Sets
A subset \(C\subseteq\mathbb{R}^n\) is convex if for every \(x,y\in C\), the entire line segment between them lies in \(C\).
That is,
\[ (1-t)x+ty\in C \]for every
\[ 0\le t\le1. \]Every convex set is path connected.
\[ \boxed{\text{convex}\Longrightarrow\text{path connected}}. \]Examples of Convex Path Connected Sets
In Euclidean space, common convex sets include:
- intervals;
- open disks;
- closed disks;
- balls;
- rectangles;
- half-planes;
- line segments;
- affine subspaces.
All are path connected.
Path Connected Does Not Mean Convex
Convexity is sufficient for path connectedness but not necessary.
For example, the unit circle
\[ S^1= \{(x,y):x^2+y^2=1\} \]is not convex because the straight segment between opposite points passes through the disk rather than staying on the circle.
Yet the circle is path connected because one can travel continuously along the circle between any two points.
A Path on the Circle
One standard parametrization of the unit circle is
\[ \gamma(t)= \bigl( \cos(2\pi t), \sin(2\pi t) \bigr). \]As \(t\) moves from \(0\) to \(1\), the path travels once around the circle.
More generally, suitable arc parametrizations connect any chosen pair of points on the circle.
Path Connected Implies Connected
Every path connected space is connected.
\[ \boxed{ X\text{ path connected} \Longrightarrow X\text{ connected} }. \]This theorem is fundamental.
Why Path Connectedness Implies Connectedness
Suppose \(X\) were path connected but disconnected.
Then
\[ X=U\cup V \]would be a separation.
Choose
\[ x\in U \]and
\[ y\in V. \]Path connectedness would provide a continuous path
\[ \gamma:[0,1]\to X \]from \(x\) to \(y\).
But the connected interval \([0,1]\) cannot have a continuous image split between two separated pieces.
This contradiction shows that \(X\) must be connected.
The Converse Can Fail
Connectedness alone does not guarantee that every two points can be joined by a path.
Therefore,
\[ \boxed{ \text{connected} \centernot\Longrightarrow \text{path connected} }. \]The difference appears in spaces with sufficiently intricate local structure.
The Topologist's Sine Curve
A classical example is built from the graph
\[ y=\sin\left(\frac1x\right), \qquad 0Its closure adds the vertical segment
\[ \{0\}\times[-1,1]. \]The resulting set is connected but not path connected.
Why the Topologist's Sine Curve Matters
The example shows that being topologically "all in one piece" does not necessarily provide continuous routes between every pair of points.
Connectedness is a global separation property.
Path connectedness demands explicit continuous routes.
That extra requirement makes it stronger.
Continuous Images Preserve Connectedness
Suppose
\[ f:X\to Y \]is continuous and \(X\) is connected.
Then the image
\[ f(X) \]is connected.
\[ \boxed{ X\text{ connected and }f\text{ continuous} \Longrightarrow f(X)\text{ connected} }. \]Why Continuous Images Preserve Connectedness
A continuous map cannot tear a connected space into separated image pieces.
It may bend, stretch, compress, identify points, or collapse entire regions, but it cannot create a separation in the image that was absent from the domain.
This is one of the basic ways topology formalizes the idea that continuous deformation does not tear a space.
Continuous Images Preserve Path Connectedness
If \(X\) is path connected and
\[ f:X\to Y \]is continuous, then
\[ f(X) \]is path connected.
Indeed, if \(\gamma\) is a path in \(X\), then
\[ f\circ\gamma \]is a path in \(f(X)\).
Connectedness and the Intermediate Value Principle
The Intermediate Value Theorem is closely tied to connectedness.
If an interval \(I\subseteq\mathbb{R}\) is connected and
\[ f:I\to\mathbb{R} \]is continuous, then \(f(I)\) is connected.
Connected subsets of \(\mathbb{R}\) are intervals.
Therefore \(f(I)\) must contain every value between any two of its values.
Connected Components
A connected component of a space is a maximal connected subset.
"Maximal" means that the subset cannot be enlarged within the space while remaining connected.
Connected components divide a space into its largest connected pieces.
Example: Connected Components of a Gapped Set
Let
\[ X=(-3,-1)\cup\{0\}\cup[2,5]. \]Its connected components are
\[ (-3,-1), \] \[ \{0\}, \]and
\[ [2,5]. \]Each is connected, and none can be enlarged inside \(X\) without becoming disconnected.
Components Are Closed
Every connected component of a topological space is closed in that space.
Components need not always be open.
In some especially well-behaved spaces, components are open as well, but that is not true in complete generality.
Path Components
A path component is a maximal path connected subset.
Two points lie in the same path component exactly when there is a path in the space joining them.
Path components refine connected components.
Connected Components vs. Path Components
Every path component is connected because path connected spaces are connected.
Therefore every path component lies inside some connected component.
A connected component may contain several path components when the space is connected but not path connected.
Locally Path Connected Spaces
A space is locally path connected if every point has a neighborhood basis consisting of path connected open sets.
In such spaces, connected components and path components agree.
Many familiar spaces from Euclidean geometry are locally path connected.
Open Subsets of Euclidean Space
Open subsets of \(\mathbb{R}^n\) are locally path connected.
Therefore, for an open set
\[ U\subseteq\mathbb{R}^n, \]connectedness and path connectedness are equivalent:
\[ \boxed{ U\text{ open in }\mathbb{R}^n: \quad U\text{ connected} \iff U\text{ path connected} }. \]Union of Connected Sets with a Common Point
Suppose \(\{A_\alpha\}\) is a family of connected subsets of a space and they all share at least one common point:
\[ \bigcap_\alpha A_\alpha\ne\varnothing. \]Then their union is connected:
\[ \boxed{ \bigcup_\alpha A_\alpha \text{ is connected} }. \]Why a Common Point Holds Connected Pieces Together
If every connected set touches the same common point, then no separation can place one of those sets entirely on one side while placing the common point on the other.
The shared point acts as a topological bridge linking the pieces.
Example: Two Intersecting Disks
Let \(D_1\) and \(D_2\) be two disks in \(\mathbb{R}^2\).
Each disk is convex, so each is path connected and therefore connected.
If
\[ D_1\cap D_2\ne\varnothing, \]then
\[ D_1\cup D_2 \]is connected.
Union of Path Connected Sets with a Common Point
If a family of path connected sets all share a common point, then their union is path connected.
A path from \(x\) to \(y\) can travel from \(x\) to the common point inside one set and then from the common point to \(y\) inside another.
Closure of a Connected Set Is Connected
If \(A\) is connected, then its closure
\[ \overline{A} \]is connected.
\[ \boxed{ A\text{ connected} \Longrightarrow \overline{A}\text{ connected} }. \]Adding limit points cannot split a connected set into disconnected pieces.
Closure Need Not Preserve Path Connectedness
The corresponding statement for path connectedness is false in general.
A path connected set may have a closure that is connected but not path connected.
The topologist's sine curve provides a standard example: the oscillating graph itself is path connected, but its closure is not path connected.
Removing Points and Connectedness
Removing a point from a space may or may not disconnect it.
For example,
\[ \mathbb{R}\setminus\{0\} = (-\infty,0)\cup(0,\infty) \]is disconnected.
But
\[ \mathbb{R}^2\setminus\{(0,0)\} \]remains path connected.
Why Removing a Point Behaves Differently in Different Dimensions
On the real line, a point can block passage from one side to the other.
In the plane, a path can usually go around a missing point.
This simple contrast shows how connectedness can reveal qualitative differences between spaces of different dimensions.
Cut Points
A cut point of a connected space is a point whose removal disconnects the space.
Every interior point of a real interval is a cut point of that interval.
By contrast, the circle \(S^1\) has no cut point: removing one point leaves a space homeomorphic to an open interval, which remains connected.
Connectedness Is a Topological Property
Homeomorphic spaces have the same connectedness behavior.
If
\[ X\cong Y, \]then
\[ X\text{ is connected} \iff Y\text{ is connected}. \]The same is true for path connectedness.
Why Homeomorphisms Preserve Connectedness
A homeomorphism is a continuous bijection with continuous inverse.
Since continuous images of connected spaces are connected, a homeomorphism sends connected spaces to connected spaces.
Applying the inverse gives the converse.
Connectedness Does Not Depend on Distance
Connectedness is defined using topology rather than numerical distance.
A space may be connected even when no metric has been specified.
This makes connectedness a genuinely topological property rather than a purely geometric one.
Common Error: Thinking Connected Means "Touching"
Informal touching can be misleading.
Connectedness is determined by whether a separation exists.
Two pieces may share a limit point and become connected after that point is included, even if a drawing makes the connection look extremely thin.
Common Error: Thinking Connected Means Path Connected
In many elementary geometric examples, both properties occur together.
But in general,
\[ \text{connected} \centernot\Longrightarrow \text{path connected}. \]The distinction becomes important in more advanced topology.
Common Error: Requiring Separation Sets to Be Open in the Ambient Space
The sets in a separation must be open in the space under study.
If \(X\subseteq\mathbb{R}\), then relative openness in \(X\) is enough.
Forgetting the subspace topology can lead to incorrect conclusions.
Common Error: Assuming a Gap Must Have Positive Width
A space can be disconnected even when the missing region is only a single point.
For example,
\[ (-1,1)\setminus\{0\} \]is disconnected.
Topological separation is not measured by the numerical width of a gap.
Common Error: Assuming a Complicated Shape Is Disconnected
A set can be highly twisted or irregular and still be path connected.
Shape complexity alone does not determine connectedness.
What matters is whether a separation exists and, for path connectedness, whether continuous paths exist.
Common Error: Assuming Every Connected Set Is Convex
Convexity is much stronger than connectedness.
A circle is connected and path connected but not convex.
A curved arc is another simple path connected set that need not contain straight-line segments between all of its points.
A Reliable Procedure for Testing Connectedness
- Identify the topology on the space.
- Look for a decomposition into two disjoint nonempty relatively open sets.
- If such a decomposition exists, the space is disconnected.
- Look for nontrivial clopen subsets as an equivalent test.
- Use known theorems when available, such as "intervals are connected."
- Use continuous images or unions of intersecting connected sets to build new examples.
A Reliable Procedure for Testing Path Connectedness
- Choose arbitrary points \(x,y\) in the space.
- Try to construct a continuous path \(\gamma:[0,1]\to X\).
- Verify \(\gamma(0)=x\) and \(\gamma(1)=y\).
- Verify that \(\gamma(t)\in X\) for every \(t\in[0,1]\).
- Use straight-line paths when the set is convex.
- Use known path connected pieces and concatenate paths when helpful.
Example: A Rectangle
Consider
\[ R=[0,3]\times[-1,2]. \]The rectangle is convex.
Therefore any two points \(x,y\in R\) are joined by
\[ \gamma(t)=(1-t)x+ty. \]Hence,
\[ \boxed{R\text{ is path connected and connected}}. \]Example: An Annulus
Consider
\[ A= \{(x,y):1Nevertheless, one can move continuously around the hole.
Therefore the annulus is path connected.
Example: Two Disjoint Disks
Let \(D_1\) and \(D_2\) be disjoint disks with positive distance between them.
Then
\[ X=D_1\cup D_2 \]has two separated open pieces.
Therefore,
\[ \boxed{X\text{ is disconnected}}. \]It is also not path connected.
Example: Two Disks Meeting at One Point
Suppose two closed disks meet at exactly one point \(p\).
Each disk is path connected, and their intersection contains \(p\).
Therefore their union is path connected.
A connection through even one shared point can be enough.
Example: The Punctured Plane
Consider
\[ X=\mathbb{R}^2\setminus\{(0,0)\}. \]Although the origin is missing, any two points can be connected by a path that detours around the origin.
Therefore,
\[ \boxed{ \mathbb{R}^2\setminus\{(0,0)\} \text{ is path connected} }. \]Example: The Punctured Real Line
Compare the plane with
\[ \mathbb{R}\setminus\{0\}. \]Here the missing point separates the space into
\[ (-\infty,0) \]and
\[ (0,\infty). \]Hence,
\[ \boxed{ \mathbb{R}\setminus\{0\} \text{ is disconnected} }. \]Practice 1: Interval
Is
\[ [-2,5] \]connected? Is it path connected?
Practice 2: Gapped Set
Determine whether
\[ (-3,-1)\cup(2,6) \]is connected.
Practice 3: Remove a Point
Is
\[ [0,4]\setminus\{2\} \]connected?
Practice 4: Convex Set
Explain why the disk
\[ D=\{(x,y):x^2+y^2\le9\} \]is path connected.
Practice 5: Circle
Is
\[ S^1=\{(x,y):x^2+y^2=1\} \]path connected?
Practice 6: Implication
Complete:
\[ \text{path connected} \Longrightarrow \boxed{\phantom{\text{connected}}}. \]Practice 7: Converse
Is every connected space path connected?
If not, name the standard example introduced in this lesson.
Practice 8: Components
Find the connected components of
\[ X=(-4,-2)\cup\{0\}\cup[3,7]. \]Practice 9: Continuous Image
Suppose \(X\) is connected and
\[ f:X\to Y \]is continuous.
What can be concluded about \(f(X)\)?
Practice 10: Removing a Point
Compare:
\[ \mathbb{R}\setminus\{0\} \]and
\[ \mathbb{R}^2\setminus\{(0,0)\}. \]Which is disconnected, and which is path connected?
Practice Answers
Practice 1
The set \([-2,5]\) is an interval, so it is connected.
It is also convex, so it is path connected.
\[ \boxed{\text{connected and path connected}}. \]Practice 2
The two intervals are disjoint and relatively open in the union.
\[ \boxed{(-3,-1)\cup(2,6)\text{ is disconnected}}. \]Practice 3
Removing \(2\) gives
\[ [0,2)\cup(2,4]. \]These form two disjoint nonempty open subsets in the subspace.
\[ \boxed{[0,4]\setminus\{2\}\text{ is disconnected}}. \]Practice 4
The disk is convex. Therefore the straight-line segment between any two points remains inside the disk.
\[ \boxed{D\text{ is path connected}}. \]Practice 5
Yes. Any two points on the circle can be joined by a continuous arc along the circle.
\[ \boxed{S^1\text{ is path connected}}. \]Practice 6
\[ \boxed{\text{connected}}. \]Practice 7
No.
The closure of the topologist's sine curve is a standard example of a connected space that is not path connected.
Practice 8
The connected components are
\[ \boxed{ (-4,-2),\qquad \{0\},\qquad [3,7] }. \]Practice 9
Continuous images of connected spaces are connected.
\[ \boxed{f(X)\text{ is connected}}. \]Practice 10
The punctured real line is disconnected:
\[ \boxed{ \mathbb{R}\setminus\{0\} \text{ is disconnected} }. \]The punctured plane remains path connected:
\[ \boxed{ \mathbb{R}^2\setminus\{(0,0)\} \text{ is path connected} }. \]What to Remember
- A space is connected when it cannot be separated into two disjoint nonempty open pieces.
- A space is connected exactly when its only clopen subsets are the empty set and the whole space.
- Connected subsets of \(\mathbb{R}\) are exactly the intervals.
- A path is a continuous map \(\gamma:[0,1]\to X\).
- A space is path connected when any two points can be joined by a path.
- Every path connected space is connected.
- A connected space need not be path connected.
- Convex subsets of Euclidean space are path connected.
- Continuous images preserve connectedness and path connectedness.
- Connected components are maximal connected subsets.
- Path components are maximal path connected subsets.
The key distinction is:
\[ \boxed{ \text{connected} = \text{cannot be separated}, \qquad \text{path connected} = \text{can travel continuously between points} }. \]Where We Go Next
Connectedness describes whether a space comes apart into distinct topological pieces. Path connectedness adds the stronger requirement of continuous routes between points.
These ideas lead naturally to components, local connectedness, compactness, continuity, homeomorphisms, and qualitative classification of spaces. They also provide some of the first examples of topological properties that can distinguish spaces even when ordinary measurements are ignored.