Path Connected Topological Spaces

Path Connected Topological Spaces

Recall that a topological space $X$ is said to be disconnected if there exists open sets $A, B \subset X$ where $A, B \neq \emptyset$, $A \cap B = \emptyset$, and:

\begin{align} \quad X = A \cup B \end{align}

We said that $X$ is connected if it is not disconnected. The definition of a connected topological space works well with our intuition on how a connected topological spaces should behave most of the time, however, there is also a stronger for a connectedness called path connectedness which we define below.

Definition: Let $X$ be a topological space. Then $X$ is said to be Path Connected if for every distinct pair $x, y \in X$ there exists a continuous map $\alpha : [0, 1] \to X$ such that $\alpha(0) = x$ and $\alpha(1) = y$. Such functions are called Paths from $x$ to $y$.

For example, consider the topological subspace $(0, 1)$ of $\mathbb{R}$ (with the usual topology). For each $x, y \in \mathbb{R}$, define a function $\alpha : [0, 1] \to (0, 1)$ by:

\begin{align} \quad \alpha(t) = (1 - t)x + ty \end{align}

Then $\alpha$ is clearly continuous and $\alpha(0) = x$, and $\alpha(1) = y$. Thus $(0, 1)$ is path connected.

More generally, $\mathbb{R}^n$ is path connected. Let $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$ and define a function $\alpha : [0, 1] \to \mathbb{R}^n$ by:

\begin{align} \quad \alpha(t) = (1 - t) \mathbf{x} + t \mathbf{y} \end{align}

Then once again $\alpha$ is continuous and $\alpha(0) = \mathbf{x}$ while $\alpha(1) = \mathbf{y}$.

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