Simply Connected Space
In topology, a topological space is called simply connected (or 1-connected) if it is path-connected and every path between two points can be continuously transformed, staying within the space, into any other path while preserving the two endpoints in question (see below for an informal discussion).
If a space is not simply connected, it is convenient to measure the extent to which it fails to be simply connected; this is done by the fundamental group. Intuitively, the fundamental group measures how the holes behave on a space; if there are no holes, the fundamental group is trivial — equivalently, the space is simply connected.
Read more about Simply Connected Space: Informal Discussion, Formal Definition and Equivalent Formulations, Examples, Properties
Famous quotes containing the words simply, connected and/or space:
“The 5307th has collapsed. From a medical viewpoint, they’re finished as a fighting unit.... I have never seen human beings in such condition. They’re drained, physically and psychologically drained. I’m not tagging them for specific ailments. I’m simply marking every man in the outfit A.O.E.—accumulation of everything.”
—Samuel Fuller, U.S. screenwriter, and Milton Sperling. Samuel Fuller. Doc (Andrew Duggan)
“When, in the course of human events, it becomes necessary for one people to dissolve the political bands which have connected them with another, and to assume the powers of the earth, the separate and equal station to which the laws of nature and of nature’s God entitle them, a decent respect to the opinions of mankind requires that they should declare the causes which impel them to the separation.”
—Thomas Jefferson (1743–1826)
“The true gardener then brushes over the ground with slow and gentle hand, to liberate a space for breath round some favourite; but he is not thinking about destruction except incidentally. It is only the amateur like myself who becomes obsessed and rejoices with a sadistic pleasure in weeds that are big and bad enough to pull, and at last, almost forgetting the flowers altogether, turns into a Reformer.”
—Freya Stark (1893–1993)