Differential Geometry | Topology Answers

Let E be a Euclidian vector space, let (i,j) be its standard basis. Note C

the circle centered at the origin with radius a, where a is a real positive number. ⃗⃗

1. Let δ = (a cos as )i+(a sin as )j. Show that ([0, 2πa], δ) is a unit speed parametrization of C

Prove that the boundary of a subset A of a metric space X is always a closed set

Prove or disprove any metric defined on X(#0) induces a topology on X

Prove or disprove every topological space is metrizible

2.A stone moves along the path x=t^3+1,y=t^3, z=2t+5, where t denote time. What is the component of (dp)/dt ?

3i+2j-2k

3i-2j+2k

3i+2j+2k

-3i+2j+2k

Show that the set of rational numbers with the subspace topology of R is disconnected

Prove or disprove

1) every topological space is metrizable

2) any metric defined on X(is not equal to 0) induces a topology on X

Prove that the boundary of a subset A of a metric space X is always a closed set

Give an example of a regular space that is not normal