## What is Polygonally connected?

Definition 2.3 A polygonally connected path is the join of a collection of line segments [a1,a2],[a2,a3],…,[an-1,an] in the complex plane. Definition 2.4 A subset S of C is polygonally connected if for any a, b ∈ S there is a polygonally connected path in S with endpoints a and b.

### Is a circle Polygonally path connected?

The unit circle C is pathwise connected, and connected. It is not polygonally connected (and it is closed, not open).

#### Does connected imply path connected?

A locally path-connected space is path-connected if and only if it is connected. The closure of a connected subset is connected. Furthermore, any subset between a connected subset and its closure is connected. The connected components of a locally connected space are also open.

How do you show a set is connected?

To prove that X is connected, you must show no such A and B can ever be found – and just showing that a particular decomposition doesn’t work is not enough. Sometimes (but very rarely) open sets can be analyzed directly, for example when X is finite and the topology is given by an explicit list of open sets.

What is path connected set?

11.6 Definition A subset A of M is said to be path-connected if and only if, for all x,y ∈ A, there is a path in A from x to y. 11.7 A set A is path-connected if and only if any two points in A can be joined by an arc in A.

## What does it mean for a set to be connected?

A connected set is a set that cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set. Equivalently, it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other.

### Does contractible imply connected?

Definition 3. A space X is called simply-connected if π1(X, x) is trivial for any x ∈ X. Remark 1. So a contractible space is also simply-connected.

#### Is the union of connected sets connected?

The union of two connected sets in a space is connected if the intersection is nonempty. (Proof: Suppose that X ∩ Y has a point p in it and that X and Y are connected. If X ∪ Y is the union of disjoint sets A and B, both open in A ∪ B, then p belongs to A or B, say A.

Are open connected sets path connected?

An open set A in Rn is connected if and only if it is path- connected. Proof. Since path-connectedness implies connectedness we need to only show that A is path-connected if it is connected. Suppose A is nonempty and connected.

What do you mean by connected set?

## Is interval connected set?

If A is a non-empty set containing a finite number of points in R”, then A is disconnected. The most important example of a connected space is an interval in R, which means either an open interval, closed interval, or half-open interval. The limits can be numbers or +∞ or -∞.