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]
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.2: Vector Spaces
Problem 38E: Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1)...
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![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]
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