Connected Spaces

Introduction: what does it mean for a space to stay together? 🌉

In topology, students, one big question is whether a space is “all in one piece” or whether it can be split into separate parts. A connected space is a space that cannot be separated into two nonempty open pieces. This idea shows up in graphs, intervals on the number line, circles, and many other shapes you meet in real life and in mathematics. For example, a single piece of string is connected, while two far-apart pieces of string are not. 🧵

Learning objectives

By the end of this lesson, students, you should be able to:

explain the main ideas and terminology behind connected spaces,
apply basic topology reasoning to decide whether a space is connected,
connect the idea of connectedness to Midterm 1 and the broader topic of connectedness,
summarize why connected spaces matter in topology,
use examples and counterexamples to support your reasoning.

A key theme is that topology studies properties that do not change when a space is stretched or bent without tearing. Connectedness is one of those properties. It helps describe whether a space has “breaks” or “gaps” in a topological sense.

The definition of connectedness

A topological space $X$ is connected if there do not exist two sets $U$ and $V$ such that:

$U$ and $V$ are open in $X$,
$U \neq \varnothing$ and $V \neq \varnothing$,
$U \cap V = \varnothing$,
$U \cup V = X$.

If such a pair exists, then $X$ is disconnected. The pair $U, V$ is called a separation of $X$.

This definition may look technical, but the idea is simple: a connected space cannot be split into two open pieces that do not touch each other. Think about a class of students standing in one group versus two groups with a clear empty space between them. If there is a true gap, the set is not connected. 🧍‍♀️🧍‍♂️

There is also an equivalent way to think about connectedness using sets that are both open and closed. A subset of a space is called clopen if it is both open and closed. A space $X$ is connected if and only if the only clopen subsets of $X$ are $\varnothing$ and $X$ itself.

Why is this useful? Because it often gives a faster way to prove connectedness or disconnectedness. If you can find a nontrivial clopen set, the space is disconnected.

Examples that build intuition

Example 1: an interval in $\mathbb{R}$

The interval $[0,1]$ is connected, and so is $(0,1)$, or any interval in $\mathbb{R}$. Intuitively, you can move from one point to another without jumping over a gap. In fact, every interval in $\mathbb{R}$ is connected.

This is one of the most important examples in all of topology and analysis. Intervals are the basic “one-piece” sets on the real line.

Example 2: two separated intervals

The set $[0,1] \cup [2,3]$ is disconnected in $\mathbb{R}$ with the usual topology. Why? Because the set can be separated into the two open-in-the-subspace pieces $[0,1]$ and $[2,3]$. There is a gap between $1$ and $2$, so the set is not all in one piece.

Example 3: the circle

The circle $S^1$ is connected. Even though it curves around, it has no breaks. You can travel around it continuously. A loop of rope is a good physical model: if the rope is one continuous loop, it is connected. 🔄

Example 4: discrete spaces

If $X$ has the discrete topology and contains at least two points, then $X$ is disconnected. In a discrete topology, every subset is open. So if $X$ has two different points $a$ and $b$, then $\{a\}$ and $X \setminus \{a\}$ are both open, nonempty, disjoint, and their union is $X$. That is a separation.

These examples show that connectedness depends on the topology, not just the number of points.

How to prove a space is connected

A common method is to use known connected spaces and build from them.

Continuous images of connected spaces

If $X$ is connected and $f : X \to Y$ is continuous, then $f(X)$ is connected. This is a major theorem. It says continuous maps cannot “create” a separation out of nowhere.

For instance, if $[0,1]$ is connected and $f : [0,1] \to \mathbb{R}$ is continuous, then the image $f([0,1])$ must be connected. In $\mathbb{R}$, connected subsets are intervals, so $f([0,1])$ is an interval.

This is one reason connectedness matters in real-world modeling. If a temperature changes continuously along a rod, the set of possible temperatures along the rod must form a connected set of values. 🌡️

Unions of connected sets

If a family of connected sets overlaps in the right way, their union can be connected. A standard result says: if $A$ and $B$ are connected and $A \cap B \neq \varnothing$, then $A \cup B$ is connected.

This is intuitive: two connected pieces that touch form one connected whole. You can think of two rooms joined by an open doorway. Since there is a shared point or overlap, the whole floor plan may be connected.

More generally, the union of connected sets with a connected intersection can often be shown connected, depending on the situation.

How to prove a space is disconnected

To prove that a space is disconnected, it is enough to find a separation or a nontrivial clopen subset.

Method 1: find a separation

Suppose you can write $X = U \cup V$ where $U$ and $V$ are disjoint, nonempty, and open in the subspace topology. Then $X$ is disconnected.

For example, in $[0,1] \cup [2,3]$, let $U = [0,1]$ and $V = [2,3]$. Each is open in the subspace because each can be written as an intersection with an open set in $\mathbb{R}$, such as $( -1, 1.5 )$ and $(1.5, 4)$.

Method 2: use clopen sets

If you can find a subset $A$ of $X$ such that $A \neq \varnothing$, $A \neq X$, and $A$ is clopen in $X$, then $X$ is disconnected.

For example, in a discrete space, every subset is clopen. That makes disconnectedness easy to detect.

Method 3: use a continuous function to $\{0,1\}$

Another useful fact is that if there is a continuous function $f : X \to \{0,1\}$ that is not constant, then $X$ is disconnected, because $\{0,1\}$ has the discrete topology and is disconnected. This technique appears often in proofs.

Connectedness and path connectedness

Connectedness is related to, but not the same as, path connectedness.

A space $X$ is path connected if for any two points $x, y \in X$, there exists a continuous function $\gamma : [0,1] \to X$ such that $\gamma(0) = x$ and $\gamma(1) = y$.

Every path connected space is connected. The idea is that if you can draw a continuous path between any two points, then the space cannot be split into two separated open pieces.

However, the converse is not always true. There are connected spaces that are not path connected. This is an important midterm-level distinction. It shows that connectedness is a weaker condition than path connectedness.

A classic example is the topologist’s sine curve closure, which is connected but not path connected. You do not need to memorize every detail right away, but you should remember the key fact: connected does not always mean path connected.

Why this matters for Midterm 1

Connected spaces are often tested in Midterm 1 because the topic brings together several core ideas from topology:

open and closed sets,
subspace topology,
continuity,
examples and counterexamples,
proofs using definitions.

When studying connectedness, students, you should be comfortable moving between definitions and examples. Midterm problems may ask you to decide whether a set is connected, prove that a continuous image is connected, or explain why a space is disconnected using a separation.

A strong exam strategy is to ask:

What topology is being used?
Can I find a separation?
Can I use a known theorem about continuous images or unions?
Is there a simpler known connected set related to this one?

These questions help you organize your proof and avoid guesswork.

Conclusion

Connected spaces describe when a topological space stays in one piece. The definition uses open sets, separations, and clopen subsets, but the main idea is very visual: no topological gaps. Connectedness is central to Midterm 1 because it connects definitions, proofs, and examples in a way that tests your understanding of the whole course so far. It also prepares you for path connectedness, which is a stronger condition. If you can recognize connected spaces, prove connectedness, and prove disconnectedness with clear reasoning, you are building one of the most important skills in topology. ✅

Study Notes

A space $X$ is connected if it cannot be written as $X = U \cup V$ where $U$ and $V$ are disjoint, nonempty, open subsets of $X$.
A subset is clopen if it is both open and closed.
A space is connected if and only if its only clopen subsets are $\varnothing$ and $X$.
Every interval in $\mathbb{R}$ is connected.
A space with the discrete topology and at least two points is disconnected.
The continuous image of a connected space is connected.
If $A$ and $B$ are connected and $A \cap B \neq \varnothing$, then $A \cup B$ is connected.
Path connected means any two points can be joined by a continuous path $\gamma : [0,1] \to X$.
Every path connected space is connected, but not every connected space is path connected.
For proofs, students, always identify whether you need a separation, a clopen set, or a theorem about continuous images.