The Timpson Lyceum · General Topology (Point Set Topology)

Connected Topological Spaces – Lecture 18 (Topology)

Overview

This lecture introduces connectedness, a new topological property that, like the Hausdorff property, is preserved under homeomorphisms. Informally, a space is connected if it cannot be broken cleanly into two non-overlapping open pieces. We make this precise through the notion of a separation, classify several familiar spaces as connected or disconnected, and develop the tools needed to reason about connected subsets. The key structural results are: a space is disconnected exactly when it contains a proper non-empty clopen subset, continuous functions preserve connectedness, and connectedness survives both the inclusion of limit points and the union/product of overlapping connected sets. (A second notion, path connectedness, is previewed for later.)

Key Concepts

Separation and disconnectedness. Let $(X,\tau)$ be a topological space. We say $X$ is disconnected if and only if there exist open sets $U, V \in \tau$ such that $U \cup V = X, \qquad U \cap V = \varnothing, \qquad U \neq \varnothing, \qquad V \neq \varnothing.$ Such a pair $U, V$ is called a separation of $X$. The requirement that both sets be non-empty is essential — otherwise every space would be "disconnected" via the trivial split $X = X \cup \varnothing$.

Connectedness. $X$ is connected if and only if it is not disconnected. Unlike "open" and "closed" (a set can be both, either, or neither), every space is exactly one of connected or disconnected.

Clopen characterization. $X$ is disconnected $\iff$ there exists a proper, non-empty subset $A \subsetneq X$ that is clopen (both open and closed). The link: if $\{U,V\}$ is a separation, then $U = V^{c}$, so $U$ is open and its complement $V$ is open — making $U$ closed as well.

Examples on a 3-point set $X=\{a,b,c\}$.

• $\tau = \{\varnothing, \{b\}, \{b,c\}, \{a,b\}, X\}$ is connected: no two of these open sets partition $X$ disjointly.
• A topology in which $\{a\}$ and $\{c\}$ are both open is disconnected: $\{a\}$ and $\{c\}$ together with appropriate sets give a separation.

Standard spaces.

• $\mathbb{R}$ with the standard topology is connected. Any attempted split, e.g. $U=(-\infty,0)$ and $V=(0,\infty)$, misses the point $0$; assigning $0$ to either side forces that side to be a half-open (non-open) set.
• The subspace $(-1,0)\cup(0,1)$ of $\mathbb{R}$ is disconnected: each interval is open in the subspace, they are disjoint, non-empty, and union to the whole set.
• $\mathbb{R}\setminus\{p\}$ (delete one point) is disconnected: $(-\infty,p)$ and $(p,\infty)$ form a separation.
• The discrete topology $(X, \mathcal{P}(X))$ with $|X|\geq 2$ is disconnected: pick any proper non-empty $A\subsetneq X$; then $A$ and $A^{c}$ are both open and form a separation.
• The trivial (indiscrete) topology $\{\varnothing, X\}$ is connected — there aren't even two non-empty open sets to separate.
• $\mathbb{R}$ with the lower-limit topology (basis $[a,b)$) is disconnected: for any point, the ray on each side is a union of basis elements, hence clopen.

Connected subsets — two equivalent viewpoints. For $A \subseteq X$:

• $A$ is connected in $X$ if the subspace topology on $A$ is connected.
• A separation of $A$ in $X$ is a pair $U,V$ open in $X$ with $A \subseteq U\cup V$, $U\cap A \neq \varnothing$, $V\cap A \neq \varnothing$, and $U\cap V\cap A = \varnothing$. (Note $U,V$ may overlap, but not inside $A$.) A theorem establishes these are logically equivalent: $A$ has a separation in $X$ iff $A$ is disconnected as a subspace, via $U' = U\cap A$, $V' = V\cap A$.

Major theorems.

• Continuity preserves connectedness: if $X$ is connected and $f:X\to Y$ is continuous, then $f(X)$ is connected in $Y$.
• Connected sets stay on one side: if $C\subseteq D\subseteq X$ with $C$ connected, then any separation $\{U,V\}$ of $D$ has $C\subseteq U$ or $C\subseteq V$ (never split across both).
• Closure preserves connectedness: if $C$ is connected and $C\subseteq A\subseteq \overline{C}$, then $A$ is connected. (Adding limit points cannot disconnect a set.)
• Union with a common point: if $\{C_\alpha\}$ is a collection of connected sets with $\bigcap_\alpha C_\alpha \neq \varnothing$, then $\bigcup_\alpha C_\alpha$ is connected.
• Products: if $X$ and $Y$ are connected, then $X\times Y$ is connected.

Worked Example

Claim. Continuous functions preserve connectedness: if $X$ is connected and $f:X\to Y$ is continuous, then $f(X)$ is connected in $Y$.

Proof (by contradiction). Suppose $f(X)$ is disconnected in $Y$. Then there exist open sets $V_1, V_2$ in $Y$ that separate $f(X)$. Take the preimages $U_1 = f^{-1}(V_1)$, $U_2 = f^{-1}(V_2)$, which are open in $X$ by continuity. We verify these separate $X$:

Thus $\{U_1, U_2\}$ is a separation of $X$, so $X$ is disconnected — contradicting the hypothesis that $X$ is connected. Therefore $f(X)$ must be connected. $\blacksquare$

Summary

• A space is disconnected when it has a separation — two disjoint, non-empty open sets covering it; it is connected when no such split exists, and every space is exactly one of the two.
• Equivalently, $X$ is disconnected $\iff$ it contains a proper, non-empty clopen subset.
• Familiar verdicts: $\mathbb{R}$ (standard) is connected; deleting a point, the discrete topology, and the lower-limit topology are all disconnected; the trivial topology is connected.
• Continuous images of connected spaces are connected, proved by pulling a hypothetical separation of $f(X)$ back through $f^{-1}$.
• Connectedness is robust: a connected subset always lands entirely inside one side of any separation, it survives passage to its closure, and unions of connected sets sharing a common point (as well as products of connected spaces) remain connected.
• These results build toward components (maximal connected pieces), defined later via an equivalence relation.