Correct my classification.

Question

This figure is taken from Wolfram MathWorld. The site classified the third region as not simply connected, and I get it, but it isn't also multiply connected, since multiply connected means that I can connect any two points in the region with a curve that lies entirely in the region. However, it is clear that I cant connect one point in the upper subregion with another one in the down subregion with a curve that lies totally in the region, correct?

Note

A multiply connected region is a connected region but not simply.

Answer 1

Often times, a plane region $D$ is simply-connected if

• Every closed path in $D$ is homotopic in $D$ to a constant path, and
• $D$ is connected. (For at least some authors, a region is a non-empty, connected open set, so connectedness may be left unstated in the definition of simple-connectedness.)

In my experience, one says $D$ is multiply-connected if there exists a closed path in $D$ that is not homotopic in $D$ to a constant path, i.e., given two points of $D$ there exist multiple non-homotopic paths joining them. Offhand I expect (but don't know for certain whether) one normally speaks only of "multiply-connected regions," i.e., not of "multiply-connected (open) sets."

---

The third set is not connected, hence not simply-connected. Every closed path in $D$ is homotopic in $D$ to a constant path, however, so (using the criterion above, whether or not connectedness is assumed) the third set is also not multiply-connected.

Answer 2

Multiply connected is not a known term, so you can't use it without providing a definition. Usually the definition of simply connected (any two paths joining points $p$ and $q$ are homotopic) assumes the space is path connected, which avoids the difficulty of the third object. Similar considerations if you define simply connected as having trivial fundamental group (any based loop is null homotopic). By that definition, the third object is simply connected for any basepoint, but what about a space that is the disjoint union of one of the "blobs" in the third figure with the fourth figure? Now simply connected or not depends on which path component you are talking about. These issues are all surmountable just by using clear definitions, and the simplest conclusion to your situation is that the space in the third picture is not path connected, but each path component is simply connected.