TY - JOUR
T1 - There is a continuum which is connected by uniformly short paths but not uniformly path connected
AU - Holmes, Melvin R.
PY - 1991/11/12
Y1 - 1991/11/12
N2 - David Bellamy defined the concept of connectedness by uniformly short paths and asked wether it was equivalent to the notion of uniform path connectedness. A counterexample is described. Some definitions: for a positive real number e, a Peano continuum has "e-length" ≤ an integer n iff it is the union of a chain of n Peano continua of diameter less than e. A continuum (compact connected metric space) X is "uniformly path connected" iff there is a family P of paths such that for each a, b in X, there is a path in P from a to b, and for each e there is an N such that the e-length of the image of each path in P is ≤ N. A continuum is "connected by uniformly short paths" iff for each e there is a number N such that for each pair of points a, b in X there is a path from a to b whose image has e-length ≤ N. (The definition of e-length given is more concise than that given in the paper).
AB - David Bellamy defined the concept of connectedness by uniformly short paths and asked wether it was equivalent to the notion of uniform path connectedness. A counterexample is described. Some definitions: for a positive real number e, a Peano continuum has "e-length" ≤ an integer n iff it is the union of a chain of n Peano continua of diameter less than e. A continuum (compact connected metric space) X is "uniformly path connected" iff there is a family P of paths such that for each a, b in X, there is a path in P from a to b, and for each e there is an N such that the e-length of the image of each path in P is ≤ N. A continuum is "connected by uniformly short paths" iff for each e there is a number N such that for each pair of points a, b in X there is a path from a to b whose image has e-length ≤ N. (The definition of e-length given is more concise than that given in the paper).
KW - Connected
KW - connected by uniformly short paths
KW - continuum
KW - path connected
KW - uniformly path connected
UR - http://www.scopus.com/inward/record.url?scp=15944416043&partnerID=8YFLogxK
U2 - 10.1016/0166-8641(91)90029-L
DO - 10.1016/0166-8641(91)90029-L
M3 - Article
AN - SCOPUS:15944416043
SN - 0166-8641
VL - 42
SP - 17
EP - 23
JO - Topology and its Applications
JF - Topology and its Applications
IS - 1
ER -