Differential Geometry | Topology Answers

Show that the fundamental group π1(S^1) is isomorphic to the additive group Z of integers .

Show that the fundamental group π1(S^1) is isomorphic to the additive group Z of integers .

Prove that an infinite product of discrete spaces may not be discrete.

Prove
1. If X is connected, then every quotient space of X is connect .
2. If X is compact ,then every quotient of X is compact.

Prove that a space X is homeomorphic to an open subspace of a compact Hausdorff space if and only if X is locally compact.

Show that Hilbert space is not locally compact at any point.

Find the unit tangent vector at the indicated point of the vector function
r(t)=e^(12t)cost i+e^(12t)sint j+e^(12t) k

T(π/2)=<_,_,_>

Bob claims that he can map Rectangle 1 to Rectangle 2.
Select THREE of the transformations or series of transformations that support his claim.
A
a translation of 2 units down followed by a 180° rotation about the point (1, 2)

B
a reflection over the line x = 0 followed by a reflection over the line y = 4

C
a reflection over the line x = 1 followed by a translation of 2 units down

D
a translation of 12 units right and 2 units down

E
a rotation of 180° about the point (1, 4)

F
a translation of 6 units right

Determine the unit tangent vector at the point (2,4,7) for the circle with parametric equations x=2u, y=u^2 +3 and z=2u^2 +5