Real Number Line with Point Removed is Not Path-Connected
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Theorem
Let $\R$ be the real number line considered as an Euclidean space.
Let $x \in \R$ be a real number.
Then $\R \setminus \set x$, where $\setminus$ denotes set difference, is not path-connected.
Proof
We have that $x - 1$ and $x + 1$ are both real numbers, so:
- $x - 1 \in \R \setminus \set x$
- $x + 1 \in \R \setminus \set x$
Let $\mathbb I := \closedint 0 1$ be the closed unit interval.
Suppose there exists a path $f: \mathbb I \to \R \setminus \set x$ from $x - 1$ to $x + 1$.
Then by Image of Interval by Continuous Function is Interval, it follows that $x = \map f x$ for some $s \in \mathbb I$.
But $x \notin \R \setminus \set x$ by definition of set difference.
Hence such an $f$ can not exist.
The result follows by definition of path-connected.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Path-Connectedness