Exercise 1. 1) Suppose the topological space X is the union of the finite family F₁,..., Fm of closed subsets of X. If f: X→Y is a map from X to the topological space Y, show that f: X→Y is continuous if and only if each restriction f|F₁: F₁ →Y, i = 1,...,m, is continuous. 2) If we assume that X = UiENFi, with each F₁ a closed subset of the topological space X, and f: X → Y is a map such that every restriction fF; : F; → Y, i € N, is continuous, does it follow that f is automatically continuous? [If true prove it, if false give an example]
Exercise 1. 1) Suppose the topological space X is the union of the finite family F₁,..., Fm of closed subsets of X. If f: X→Y is a map from X to the topological space Y, show that f: X→Y is continuous if and only if each restriction f|F₁: F₁ →Y, i = 1,...,m, is continuous. 2) If we assume that X = UiENFi, with each F₁ a closed subset of the topological space X, and f: X → Y is a map such that every restriction fF; : F; → Y, i € N, is continuous, does it follow that f is automatically continuous? [If true prove it, if false give an example]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Exercise 1
Need part 1 and part 2
![Exercise 1. 1) Suppose the topological space X is the union of the finite family F₁,.
subsets of X.
Fm of closed
If f: X→Y is a map from X to the topological space Y, show that f: X→Y is continuous if
and only if each restriction f|F: F₁ →Y, i = 1, ..., m, is continuous.
2) If we assume that X = UieNFi, with each F; a closed subset of the topological space X, and
f: X→ Y is a map such that every restriction f F₁ : F; → Y, i ≤ N, is continuous, does it follow
that f is automatically continuous? [If true prove it, if false give an example]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa68164dd-6bba-4aa5-92bc-4824a71db092%2F3f167378-34b4-4a06-b3d1-9fef7c709894%2Fbx72fqv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 1. 1) Suppose the topological space X is the union of the finite family F₁,.
subsets of X.
Fm of closed
If f: X→Y is a map from X to the topological space Y, show that f: X→Y is continuous if
and only if each restriction f|F: F₁ →Y, i = 1, ..., m, is continuous.
2) If we assume that X = UieNFi, with each F; a closed subset of the topological space X, and
f: X→ Y is a map such that every restriction f F₁ : F; → Y, i ≤ N, is continuous, does it follow
that f is automatically continuous? [If true prove it, if false give an example]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 34 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
