In his book "Geometry, Topology and Physics", Nakahara makes the following claim with regard to topological spaces:
With a few pathological exceptions, arcwise connectedness is practically equivalent to connectedness.
Could somebody please give me examples of such exceptions? Where can I read more about these pathologies?
To make my question more precise: are there topological spaces which are arcwise-connected, but not connected?
On
As others have pointed out, every arcwise connected space is connected.
$\pi$-Base, an online version of the general reference chart from Steen and Seebach's Counterexamples in Topology, gives the following examples of spaces that are connected but not arcwise connected. You can learn more about these spaces by viewing the search result.
A Pseudo-Arc
An Altered Long Line
Cantor’s Leaky Tent
Closed Topologist’s Sine Curve
Countable Complement Extension Topology
Countable Complement Topology
Countable Excluded Point Topology
Countable Particular Point Topology
Divisor Topology
Double Pointed Countable Complement Topology
Finite Complement Topology on a Countable Space
Finite Excluded Point Topology
Finite Particular Point Topology
Gustin’s Sequence Space
Indiscrete Irrational Extension of the Reals
Indiscrete Rational Extension of the Reals
Irrational Slope Topology
Lexicographic Ordering on the Unit Square
Nested Angles
One Point Compactification fo the Rationals
Pointed Irrational Extension of the Reals
Pointed Rational Extension of the Reals
Prime Ideal Topology
Prime Integer Topology
Relatively Prime Integer Topology
Roy’s Lattice Space
Sierpinski Space
Smirnov’s Deleted Sequence Topoogy
The Extended Long Line
The Infinite Broom
The Infinite Cage
The Integer Broom
Topologist’s Sine Curve
Uncountable Excluded Point Topology
Uncountable Particular Point Topology
Arcwise implies path connected, which implies connected. It's the other way that has counterexamples, such as topologist's sine curve.