Let X be a topological space with the topology τ, and let A⊂X be a subset. Consider the subspace topology on A, denoted by τ_A, which is defined as τ_A={U∩A∣U∈τ}. Suppose X=R with the standard topology (where the open sets are unions of open intervals), and A=Q, the set of all rational numbers. Define a function f:A→R by f(x)=x^2. Determine whether f is continuous when A is considered with the subspace topology induced from R. Justify your answer by using the definition of continuity in topological spaces, which states that a function f:(X,τ_X)→(Y,τ_Y) is continuous if for every open set V∈τ_Y, the preimage f^{−1}(V) is an open set in τ_X.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question

Let X be a topological space with the topology τ, and let A⊂X be a subset. Consider the subspace topology on A, denoted by τ_A, which is defined as τ_A={U∩A∣U∈τ}.
Suppose X=R with the standard topology (where the open sets are unions of open intervals), and A=Q, the set of all rational numbers. Define a function f:A→R by f(x)=x^2.
Determine whether f is continuous when A is considered with the subspace topology induced from R. Justify your answer by using the definition of continuity in topological spaces, which states that a function f:(X,τ_X)→(Y,τ_Y) is continuous if for every open set V∈τ_Y, the preimage f^{−1}(V) is an open set in τ_X.

Expert Solution
steps

Step by step

Solved in 3 steps

Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,