Can you think of any spaces that are connected but not path connected apart from the Topologist's Sine Curve?
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Eric Wofsey
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rk101
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I like this one: https://math.stackexchange.com/a/463753/472818 – mr_e_man May 03 '23 at 22:33
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Here are a whole bunch from $\pi$-Base, a searchable version of Steen and Seebach's Counterexamples in Topology. You can visit the search result to learn more about any of these spaces.
An Altered Long Line
A Pseudo-Arc
Cantor's Leaky Tent
Closed Topologist's Sine Curve
Countable Complement Extension Topology
Countable Complement Topology
Double Pointed Countable Complement Topology
Finite Complement Topology on a Countable Space
Gustin's Sequence Space
Indiscrete Irrational Extension of $\mathbb{R}$
Indiscrete Rational Extension of $\mathbb{R}$
Irrational Slope Topology
Lexicographic Ordering on the Unit Square
Nested Angles
One Point Compactification of the Rationals
Pointed Irrational Extension of $\mathbb{R}$
Pointed Rational Extension of $\mathbb{R}$
Relatively Prime Integer Topology
Roy's Lattice Space
Smirnov's Deleted Sequence Topology
The Extended Long Line
The Infinite Broom
The Infinite Cage
Topologist's Sine Curve
Steven Clontz
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Austin Mohr
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An example of a connected space that is not path-connected is the deleted comb space: $$ (\{0\} \times \{0,1\}) \cup (K \times [0,1]) \cup ([0,1] \times \{0\})$$ where $K = \{ \frac{1}{n} \mid n \in \mathbb{N} \}$
Taken from here.
Chris Eagle
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Rudy the Reindeer
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Another standard example is the extended long line. Counterexamples in Topology will have more, but my copy isn't to hand right now.
Chris Eagle
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The canonical example is the extended long line. You can think of the regular line $[0,\infty)$ as the product of $[0,1)$ and $\omega$ in the dictionary order topology—effectively, a countable number of copies of $[0, 1)$ pasted end-to-end.
The long line is the same way, except that instead of a countable number of copies you use an uncountable number of copies: take $[0, 1)\times\omega_1$ in the dictionary order topology, where $\omega_1$ is the smallest uncountable ordinal. Then to get the extended long line, you add one more point $p$ onto the far end. It's clearly connected, but it isn't path-connected because the path from any finite point, say $(1/2, 1)$, is too far from $p$ for the path between them to be the image of $[0,1]$.
The book Counterexamples in Topology by Seebach and Steen is good for answering questions like this.
MJD
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The long line is path connected. All locally path connected and connected spaces are path connected. – George Lowther Mar 26 '12 at 20:47
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1@George: The space I described is not locally path-connected in a neighborhood of the extra point $p$. – MJD Mar 26 '12 at 20:49
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I think I misunderstood what you meant. Shouldn't this be called the extended long line (or ray)? I don't think the long line has an extra point added at the end. – George Lowther Mar 26 '12 at 20:56
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@George: That is my fault. I will edit the reply. Thanks for pointing out that I misspoke. – MJD Mar 26 '12 at 20:58
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@George: Are you sure that the terminology on this does not vary? I know Steen and Seebach call the larger space the "extended" long line, but I remember being surprised when I first saw that, and it seems that Brett Frankel may also have been using the same different terminology that I was. I thought the example appeared in Kelley (1955) but I have only the 1975 edition online, which does not mention it by name. – MJD Mar 26 '12 at 21:10
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No, I'm not sure that the terminology doesn't vary. I did check the wikipedia link though, and they use "long line" to mean without the additional point added. – George Lowther Mar 26 '12 at 21:48
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I think Wikipedia follows S&S. 1955 Kelley is the same as 1975 Kelley. I wonder if the example is in Munkres? – MJD Mar 26 '12 at 22:50
