Purchase Solution

Open set whose preimage under the given function is not open

Not what you're looking for?

Ask Custom Question

Let g: R -> R by:

g(x) = { x^2 + 2 ... if x <= 1, 5 - x ... if x > 1 }

Find an open subset of R (w/ respect to the usual topology) whose preimage under g is not an open subset of R.

Purchase this Solution
Solution Summary

An example of an open subset of R whose preimage under the given function g is not open is provided, along with a complete, detailed justification that the preimage of that open set is not open.

Solution Preview

Consider the set S = {x: 2 < x < 3.6}. Note that S is an open subset of R with respect to the usual topology. We will prove that the preimage of S under the given function g is not an open subset of R.

First, note the following:

g(0) = 0^2 + 2 = 2, so 0 is not in the preimage of S.

g(1) = 1^2 + 2 = 3, so 1 is in the preimage of S.

2 < g(x) < 3 for every x such that 0 < x < 1

Thus at this point we know that the preimage of S contains at least the half-open interval ...

Solution provided by:
Georgia Martin, PhD
Education
  • AB, Hood College
  • PhD, The Catholic University of America
  • PhD, The University of Maryland at College Park
Recent Feedback
  • "Thanks for your assistance. "
  • "Thank you. I understand now."
  • "Super - Thank You"
  • "Very clear. I appreciate your help. Thank you."
  • "Great. thank you so much!"
Purchase this Solution
Free BrainMass Quizzes
Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

View More Free Quizzes
Related BrainMass Solutions

  • Identifying submanifolds of R^n

    One way is to exhibit the set as the preimage of a continuous map from R^n to R whose derivative map is nonzero on the level set. (This is a special case of a more general theorem.)

  • Solve: Topological Space

    Then, for every y∈V there's a number r>0 such that the set B(y,r)={z∈R: |z − y| < r} is contained in V. Show that the pre-image f^−1(V) of V under f(x) = ∥x∥ is open in X.

  • Covering Spaces : Compact Hausdorff Spaces and Homomorphisms

    Taking complements, the image of open sets is open. Thus the function is bicontinuous. Proof:2 Let f: X  Y be continuous . Let X be compact. Let B be a covering of the set f(X) by sets open in Y.

  • Open Sets, Connectivity and Continuous Functions

    Thus we have which is also an open set. Since each open set in is the union of open sets of form like , then is the union of open sets of form like , which is also an open set. Therefore, is continuous.

  • Show that f is proper if and only if f* is continuous

    Then (referring to the definition of open sets above) is an open set in for which is not open, and thus is not continuous. So f not proper implies not continuous. The contrapositive result gives continuous implies f proper.

  • What is Open Source Database and Why Use It?

    A) Before looking into what a open source database is, we can see what a open source software is in general: Open source software is computer software whose source code is available under a license (or arrangement such as the public domain) that permits

  • Real Analysis : Baire Category Theorem

    In particular, we will need the theorem that says a closed subspace of a compact space is compact. As before, we are given o1 o2 o3 etc, a countable collection of dense open sets. Let w be an arbitrary open set. Let x1 be a point in wà?

  • Algebra

    We have the following facts about a quadratic function f(x)=ax^2+bx+c. (1) If a>0, it has minimum point at x=-b/2a, the vertex is at (-b/2a, f(-b/2a)). The parabola is open upward.

  • Implicit Function Theorem

    297231 Implicit Function Theorem: Nonempty Open Set V Determine whether the equation xyz + sin(x + y + z) = 0 can be solved for z in a nonempty, open set V containing the point (x, y, z) = (0, 0, 0).

View More Related Solutions