I was reading about end compactification, following up my interest in compactification of the real line, Homeomorphism of a compact real line to the real line.

Question 1&2: Is a Warsaw circle path connected? Is it simply connected?

I was reading an article that said,the Warsaw circle is path connected, the topologist's sine circle is path-connected but it's not locally path-connected, and another article says it is not. (See http://mathforum.org/kb/message.jspa?messageID=4032108. See also: http://sci.tech-archive.net/Archive/sci.math/2005-10/msg02190.html)

So, now I am confused. The two end points of the topologist's sine curve are connected, so the Warsaw circle should be path connected. However, it should not be simply connected. Did I miss something here? Thanks.

Question 3: Is the one point compactification of the real line simply connected?

The above disagreement between articles got me thinking of a one-point compactification of a the real-line. The real line is locally connected and path connected and simply connected, correct? However, it's one point compactification is homeomorphic to a circle and a circle is not simply-connected in two dimensions. (That is, there is no way to contract the path to a point, though there is for the real line.)

Does this mean that the one-point compactification of the real line is not simply connected even though the real line is? (I was reading: https://en.wikipedia.org/wiki/Simply_connected_space)

  • $\begingroup$ Why path connected should imply simply connected? $\endgroup$
    – egreg
    Oct 14 '15 at 18:16
  • $\begingroup$ Could you please state as clearly and succintly as possible your question? If you feel like the comments you made and how you got to the problem are useful informations, can you just make a clear separation from them and the question itself? $\endgroup$
    – Aloizio Macedo
    Oct 14 '15 at 18:21
  • $\begingroup$ I wasn't saying path connected implies simply connected. Rather I was saying it is necessary if not sufficient. $\endgroup$ Oct 14 '15 at 18:40
  • $\begingroup$ Aloizio, I was trying to do that. OK. Let me edit. $\endgroup$ Oct 14 '15 at 18:40

There are two different spaces called the Warsaw circle: (1) https://en.wikipedia.org/wiki/Continuum_(topology)#/media/File:Warsaw_Circle.png and (2) http://ncatlab.org/nlab/files/warsaw.pdf.

Basically, in (1), the 'special' point - which I'll call "$*$" - is only badly behaved on one side: the circle is smooth immediately to the left of $*$, and chaotic immediately to the right. This version is indeed path connected but not simply connected.

In (2), though, $*$ is bad on both sides. (2) is thus not path connected, since there's no path connecting $*$ to any other point (in (1), such a path can be found by going around the left).

  • $\begingroup$ Thank you. I don't believe this was mentioned in the discussion I found on the web. That clears it up. Thanks for the links. $\endgroup$ Oct 15 '15 at 0:46

The one point compactification of the real line is the circle. (If you remove one point of a circle, you obtain an open interval which diffeomorphic to the real line).

  • $\begingroup$ Yes. So, by adding the one point, that is creating a one-point compactification of the real line, the real line is no longer simply connected? $\endgroup$ Oct 14 '15 at 18:41
  • $\begingroup$ @MarkLaPolla That is correct. $\endgroup$ Oct 14 '15 at 19:32
  • $\begingroup$ Thanks @NoahSchweber. And a Warsaw circle is path connected but not simply connected? $\endgroup$ Oct 15 '15 at 0:28
  • $\begingroup$ @MarkLaPolla See my answer below. $\endgroup$ Oct 15 '15 at 0:42

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