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.)

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Let R be the relation on Q defined by a/b R c/d iff ad=bc. Show that...

Let R be the relation on Q defined by a/b R c/d iff ad=bc. Show that R is an equivalence relation. Describe the elements of the equivalence class of 2/3.

Let A ⊂ R be a nonempty discrete set a. Show that A is at most...

Let A ⊂ R be a nonempty discrete set a. Show that A is at most countable b. Let f: A →R be any function, and let p ∈ A be any point. Show that f is continuous at p

Let A be a closed subset of a T4 topological space. Show that A with the...

Let A be a closed subset of a T4 topological space. Show that A with the relative topology is a normal T1

( X, τ ) is normal if and only if for each closed subset C of...

( X, τ ) is normal if and only if for each closed subset C of X and each open set U such that C ⊆ U, there exists an open set V satisfies C ⊆ V ⊆clu( V) ⊆ U

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

2. Define a relation R on pairs of real numbers as follows: (a, b)R(c, d) iff...

2. Define a relation R on pairs of real numbers as follows: (a, b)R(c, d) iff either a < c or both a = c and b ≤ d. Is R a partial order? Why or why not? If R is a partial order, draw a diagram of some of its elements. 3. Define a relation R on integers as follows: mRn iff m + n is even. Is R a partial order? Why or why not? If R is...

Let (F, <) be an ordered field, let S be a nonempty subset of F, let...

Let (F, <) be an ordered field, let S be a nonempty subset of F, let c ∈ F, and for purposes of this problem let cS = {cx | x ∈ S}. (Do not use this notation outside this problem without defining what you mean by the notation.) Assume that c > 0. (i) Show that an element b ∈ F is an upper bound for S if and only if cb is an upper bound for cS. (ii)...

Suppose K is a nonempty compact subset of a metric space X and x ∈ X....

Suppose K is a nonempty compact subset of a metric space X and x ∈ X. (i) Give an example of an x ∈ X for which there exists distinct points p, r ∈ K such that, for all q ∈ K, d(p, x) = d(r, x) ≤ d(q, x). (ii) Show, there is a point p ∈ K such that, for all other q ∈ K, d(p, x) ≤ d(q, x). [Suggestion: As a start, let S = {d(x,...

Suppose A and B are closed subsets of R. Show that A ∩ B and A...

Suppose A and B are closed subsets of R. Show that A ∩ B and A ∪ B are closed.

Let S ⊆ R be a nonempty compact set and p ∈ R. Prove that there...

Let S ⊆ R be a nonempty compact set and p ∈ R. Prove that there exists a point x_0 ∈ S which is “closest” to p. That is, prove that there exists x0 ∈ S such that |x_0 − p| is minimal.

Subjects