Connected Space is not necessarily Path-Connected

Theorem

Let $T$ be a topological space which is connected.


Then it is not necessarily the case that $T$ is path-connected.


Proof

Let $T$ be the closed topologist's sine curve.

Recall the definition of Closed Topologist's Sine Curve::

Let $G = \set {\tuple {x, \sin \dfrac 1 x} : 0 < x}$, considered as a subset of the real number plane with the usual (Euclidean) topology $\struct {\R^2, \tau_d}$.

Let $J$ be the line segment joining the points $\tuple {0, -1}$ and $\tuple {0, 1}$ in $\R^2$.

Then the set $G \cup J$ is called the closed topologist's sine curve.


From Closed Topologist's Sine Curve is not Path-Connected, $T$ is path-connected.

From Closed Topologist's Sine Curve is Connected, $T$ is connected.

Hence the result.

$\blacksquare$

Also see


Sources

  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): connected (of a space)
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