Metric and metrizes the same topology

Reference no: EM13990636

1. On R2 let d((x, y), (u, v)) := ((x - u)2 + (y - v)2)1/2 (usual metric), e(x, y) := |x - u|+ |y - v|. Show that e is a metric and metrizes the same topology as d.

2. For any topological space (X, T ) and set A ⊂ X , the boundary of A is defined by ∂ A := A\int A. Show that the boundary of A is closed and is the same as the boundary of X \ A. Show that for any two sets A and B in X,∂( A ∪ B) ⊂ ∂ A ∪ ∂ B. Give an example where ∂( A ∪ B) /= ∂ A ∪ ∂ B.

Reference no: EM13990636

Questions Cloud

With the implementation of business analytics : With the implementation of business analytics, an organization will also need to implement a good information systems plan in order to collect, manage, and organize all of the data. Not only is an information system needed, but the data has to be org..

What does it cost to have that light on for a month : In 2003, the Schuylkill Generating Station near South Street on the Schuylkill River was rebuilt as an oil fired plant. It produces 175 Megawatts. What is the retail value of 1 day's production of electricity at this plant ?

Pseudometric spaces with topologies : 1. Let (X, d) and (Y, e) be pseudometric spaces with topologies Td and Te metrized by d and e respectively. Let f be a function from X into Y . Show that the following are equivalent (as stated in the first paragraph of this chapter):

What year did leonardo dicaprio : What year did Leonardo Dicaprio get nominated and what movie was it for?

Metric and metrizes the same topology : 1. On R2 let d((x, y), (u, v)) := ((x - u)2 + (y - v)2)1/2 (usual metric), e(x, y) := |x - u|+ |y - v|. Show that e is a metric and metrizes the same topology as d.

What is the direction of the child coriolis acceleration : A child at point A walks at constant rate toward the center of a merry-go-round which is rotating at constant angular velocity ω counter clockwise. What is the direction of the child's Coriolis acceleration?

Find the most recent financial statements : List and explain at least 2 specific challenges that the competitor company's managers may face when trying to improve these ratios.

Determining the same cardinality : Let A and B be two sets such that there exists a function f from A onto B and a function g from B onto A. Show, assuming AC, that A and B have the same cardinality. Hint: If h(y) ∈ f -1({y}) for all y in B, then h is 1-1 from B into A.

What price should each firm charge : Which company, firm 1 or firm 2 is more vulnerable to price war when P = $12 and why?

User Account

All Pages