Can a metric space with the following properties exist? Any examples?

1. The number of geodesic paths between any two points is infinite but countable.

2. The infimum of the geodesic path lengths between any two points is zero.

3. For any two points there is a maximum finite geodesic path length.

Can a space with the following properties exist? Any examples?

1. The number of geodesic paths between any two points is infinite but countable.

2. The infimum of the geodesic path lengths between any two points is zero.

3. For any two points there is a maximum finite geodesic path length.

Can a space with the following properties exist? Any examples?

• The number of geodesic paths between any two points is infinite but countable.

• The infimum of the geodesic path lengths between any two points is zero.

• For any two points there is a maximum finite geodesic path length.

Can a metric space with the following properties exist? Any examples?

1. The number of geodesic paths between any two points is infinite but countable.

2. The infimum of the geodesic path lengths between any two points is zero.

3. For any two points there is a maximum finite geodesic path length.