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"