Show that a subset of R is open if an only if it can be expressed...

Show that a subset of R is open if an only if it can be expressed as a union of open intervals

Solutions

Expert Solution

let A be an open set in , then every point in A is an interior point, that is, . where are open intervals in .

now and any point in A is in one of such open intervals in that is . therefore . which shows that any open set can be expressed as union of open intervals.

The converse part is trivial as open intervals are also open sets in and union of open sets in is again an open set. Therefore union of open intervals in is an open set.

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Show that any open subset of R (w. standard topology) is a countable union of open...

Show that any open subset of R (w. standard topology) is a countable union of open intervals. Please explain how to do, I only understand why it is true. What is required to fully prove this. What definitions should I be using.

Show that any open subset of R (w std. topology) is a countable union of open intervals.

Show that any open subset of R (w std. topology) is a countable union of open intervals.What is the objective of this problem and enough to show ?

The goal is to show that a nonempty subset C⊆R is closed iff there is a...

The goal is to show that a nonempty subset C⊆R is closed iff there is a continuous function g:R→R such that C=g−1(0). 1) Show the IF part. (Hint: explain why the inverse image of a closed set is closed.) 2) Show the ONLY IF part. (Hint: you may cite parts of Exercise 4.3.12 if needed.)

1) Show that if A is an open set in R and k ∈ R \...

1) Show that if A is an open set in R and k ∈ R \ {0}, then the set kA = {ka | a ∈ A} is open.

show that every function can be expressed as the sum of an even function and an...

show that every function can be expressed as the sum of an even function and an odd function and that there is only one way to do this.

Let φ : R −→ R be a continuous function and X a subset of R...

Let φ : R −→ R be a continuous function and X a subset of R with closure X' such that φ(x) = 1 for any x ∈ X. Prove that φ(x) = 1 for all x ∈ X.'

Suppose in a closed economy, the demand for loanable funds can be expressed as r =13...

Suppose in a closed economy, the demand for loanable funds can be expressed as r =13 – 0.05Q and the supply of loanable funds can be expressed as r = 0.015Q, where r is the real interest rate expressed in percentage and Q is the quantity of loanable funds. Also assume the government initially has a balanced budget. (a) What is the equilibrium real interest rate and quantity in the loanable funds market? Show your work. (10 points) (b) Suppose a government’s budget surplus...

Question