• #### Pesky Connection

From William Elliot@21:1/5 to All on Fri Apr 8 10:31:24 2016
Let (S,d) a multi-point connected metric space.
Is there a multi-point path connected subspace of S?

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• From David Cullen@21:1/5 to William Elliot on Sat Apr 9 18:24:05 2016
William Elliot <marsh@panix.com> wrote:

Let (S,d) a multi-point connected metric space.
Is there a multi-point path connected subspace of S?

Can you provide the definition of multipoint connected?

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• From William Elliot@21:1/5 to David Cullen on Sun Apr 10 05:54:36 2016
David Cullen <davecullen@gmail.com> wrote:

William Elliot <marsh@panix.com> wrote:

Let (S,d) a multi-point connected metric space.
Is there a multi-point path connected subspace of S?

Can you provide the definition of multipoint connected?

Let (S,d) a multi-point connected metric space.
Is there a multi-point path connected subspace of S?

Can you provide the definition of multipoint connected?

Multipoint mean has more than one point.
Connected means there are no clopen sets but
the empty set and the space itself.

As it turns out, Cantor's leaky tent or teepee
in Stien's "Counterexamples in Topology". is
an example negating the question.

[
Moderator's note:
Steen & Seebach
]

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• From Robin Chapman@21:1/5 to David Cullen on Sun Apr 10 05:55:41 2016
David Cullen <davecullen@gmail.com> wrote:

William Elliot <marsh@panix.com> wrote:

Let (S,d) a multi-point connected metric space.
Is there a multi-point path connected subspace of S?

Can you provide the definition of multipoint connected?

I presume that William's question amounts to the following:
does a connected metric space with more than one point
necessarily have a path-connected subset with more than
one point?

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• From David Cullen@21:1/5 to All on Sun Apr 10 05:56:51 2016
Ok if you are asking if there is an example of a metric space which is connected, and at the same time is also totally path disconnected, then
I suspect the knaster-kuratowski fan space will suffice.  It is known
to be connected, and I suspect it is totally path disconnected as well,
though I haven't done the legwork yet. Does this work?

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