Suggested space renaming: the "Long circle" -> "Half-long Circle" + Space suggestion: The Long Circle

[jtaylor-maths]

Issue description

The "Long circle" in the pi-base is misnamed: it is one-point compactification of the long ray as opposed to the two-sided long line, so is only "long on one side".

Space renaming

I propose that the "Long Circle" is renamed to the "Half-long circle". The rationale will be given after defining the new candidate for the "Long circle". For clarity, in this issue I set the following definitions:

• The "[existing] Long Circle]", or "Half-long Circle" is the one-point compactification of the long ray (what the pi-base currently refers to as the "long circle")
• The "Long Circle [candidate]" is the one-point compactification of the two-sided long line.

Space description: The Long Circle [candidate]

Space Suggestion

The Long Circle [candidate] the one-point compactification of the (two-sided) long line.

Rationale

This space occurs as a natural example by applying the one-point compactification to the two-sided long line. Notably it is distinct from the Half-Long Circle, since it is not path-connected (see below).

Relationship to other spaces and properties

The Long Circle [candidate] is locally compact, Hausdorff, connected, but not path-connected space. Path-connecteness fails for the same reason it fails for the Closed Long Ray: there is no path from 0 to the compacitification point $\infty$.

Comparison to currently existing [existing] Long Circle

The [existing] Long Circle is defined as the quotient of the Closed Long Ray identifying $0$ with the compactification point $\infty$. This is path-connected: any point can be connected by a path to $0$ since it either already is $0=\infty$ or it lies in the image of the Long Ray, which is path connected.

On the other hand, no path from a point $a\neq 0$ in the [existing] Long Circle can be connected by a path to $\infty$ going 'to the right' i.e. such a path must be contained in the Long Ray. In this sense, the [existing] Long Circle is only long to the right; it is long on only one side.

Conclusion

I propose the following:

• the [existing] Long Circle is renamed to "The half-long circle"
• The Long Circle [candidate] is added to the pi-base under the name "Long Circle", or perhaps "(two-sided) Long Circle"

[prabau]

I wouldn't say pi-base's Long line (S196) is "misnamed". Let's call your version 2-LC (two-sided long circle) for the sake of discussion. Both spaces are "locally orderable" (so locally 1-dimensional in a way) and both are "circles" in the sense that there is no cut point, but removing two points disconnects the space. In fact, that are many more non-homeomorphic such spaces that could be viewed as "long circles".

The version we chose for pi-base was chosen because is has some interesting properties that 2-LC does not have. In particular, S196 is path connected but not locally path connected (2LC is not even path connected). S196 is also not well-based (2-LC is well-based). In a way, S196 is more interesting that 2-LC as it illustrates a more surprising combination of properties.

We chose the name "long circle" for S196 because it was mentioned in https://math.stackexchange.com/a/1499512. It has also been mentioned variously in other mathse posts. The same name for S196 has also been used in some paper in the literature: https://zbmath.org/1271.54065 (Espinoza, Gartside, Mamatelashvili 2013).

For these reasons, I think we should keep the current name for S196.
If we were to add 2-LC in the future, we can come up with a suitable name for it at that time.

[jtaylor-maths]

I hadn't realised that there are many other such "long circles" than S196 and 2-LC, that's a good point.

I haven't seen other discussion of 2-LC (other than what I've brought up with colleagues over coffee) so I'm not sure how relevant it is as an example. One property I forgot to mention is this: it is an example of a one-point compactification of a path connected space which fails to be path-connected, perhaps this is of interest.