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?

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    $\begingroup$ think of the set $\overline{\{(x,\sin (1/x)) | x\in(0,2\pi]\}}$ this set is connected but not pathwise connected $\endgroup$ – Quickbeam2k1 Jun 22 '14 at 20:06
  • $\begingroup$ @Quickbeam2k1: I know of this example, but I meant it the other way around. $\endgroup$ – Frederic Brünner Jun 22 '14 at 20:09
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    $\begingroup$ Arc connectedness $\Rightarrow$ Path connectedness $\Rightarrow$ Connectedness, so the example you are looking for doesn't exist. If you are interested in pathological topological spaces I suggest you the book: "Steen and Seebach - Counterexamples in topology" $\endgroup$ – Dario Jun 22 '14 at 20:27
  • $\begingroup$ @Dario: Thank you for the recommendation, I will take a look at it! $\endgroup$ – Frederic Brünner Jun 22 '14 at 20:34
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    $\begingroup$ I think maybe you're misunderstanding the term is equivalent to. To say that property X is equivalent to property Y is to say that X implies Y and Y implies X. An exception to either implication is an exception to the equivalence. In your example, there are no exceptions to "arcwise connectedness implies connected", but there are pathological exceptions to "arcwise connectedness is equivalent to connectedness". $\endgroup$ – ruakh Jun 23 '14 at 0:39

Arcwise implies path connected, which implies connected. It's the other way that has counterexamples, such as topologist's sine curve.

  • $\begingroup$ Is this really always true? I thought that the word "pathological" might indicate that there are some strange counterexamples to this. $\endgroup$ – Frederic Brünner Jun 22 '14 at 20:10
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    $\begingroup$ @FredericBrünner Path-connected always implies connected. There are no counterexamples to it. The proof is an easy exercise. See this. $\endgroup$ – Ayman Hourieh Jun 22 '14 at 20:37
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    $\begingroup$ @FredericBrünner: In the context of the quote you give, I will venture that the author of that quote would consider the topologist's sine curve to be "pathological". $\endgroup$ – Lee Mosher Jun 22 '14 at 23:03
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    $\begingroup$ Nb. arcwise connected is equivalent to path connected whenever the space in question is Hausdorff, so (according to most people), counterexamples to this equivalence are very pathological (the simplest one is probably the interval with a doubled endpoint). $\endgroup$ – tomasz Jun 22 '14 at 23:52

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


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