Sentence examples for path connected from inspiring English sources

Let be a map between connected, locally path connected, and semilocally simply connected spaces.

Let be a continuous map between Hausdorff, normal, connected, locally path connected, and semilocally simply connected spaces, and let be a given base point.

We answer completely this question for the cases in which is a connected, locally path connected and semilocally simply connected space, and and are manifolds either compact or triangulable.

The Bernhard and Lehman houses were unusual in that a path connected their two yards.

Since, is path connected.

Since an absolute retract is path connected, we see that Θ is path connected.

Since an absolute retract is path-connected, Θ is path-connected.

However, in contrast to contractible sets, a compact (R_{delta}) set D need not be path-connected.

Compare with the fact that H1(X) is the abelianization of the fundamental group π1 X) when X is path-connected.

Whenever A∩B is path-connected the reduced Mayer Vietoris sequence yields the isomorphism H_1 X) \cong (H_1(A)\oplus H_1(B))/\text{Ker} (k_* - l_*) where, by exactness, :\text{Ker} (k_* - l_*) \cong \text{Im} (i_*, j_*).

Thus (P(Theta)) is path-connected in (C overline{Omega})). Since (P(Theta)=S), we see that S is nonempty and path-connected in (C overline{Omega})).