Book contents
- Frontmatter
- INTRODUCTION
- Contents
- CHAPTER 1 COMBINATORIAL GROUP THEORY
- CHAPTER 2 SPACES AND THEIR PATHS
- CHAPTER 3 GROUPOIDS
- CHAPTER 4 THE FUNDAMENTAL GROUPOID AND THE FUNDAMENTAL GROUP
- CHAPTER 5 COMPLEXES
- CHAPTER 6 COVERINGS OF SPACES AND COMPLEXES
- CHAPTER 7 COVERINGS AND GROUP THEORY
- CHAPTER 8 BASS-SERRE THEORY
- CHAPTER 9 DECISION PROBLEMS
- CHAPTER 10 FURTHER TOPICS
- NOTES AND REFERENCES
- BIBLIOGRAPHY
- INDEX
CHAPTER 2 - SPACES AND THEIR PATHS
Published online by Cambridge University Press: 08 January 2010
- Frontmatter
- INTRODUCTION
- Contents
- CHAPTER 1 COMBINATORIAL GROUP THEORY
- CHAPTER 2 SPACES AND THEIR PATHS
- CHAPTER 3 GROUPOIDS
- CHAPTER 4 THE FUNDAMENTAL GROUPOID AND THE FUNDAMENTAL GROUP
- CHAPTER 5 COMPLEXES
- CHAPTER 6 COVERINGS OF SPACES AND COMPLEXES
- CHAPTER 7 COVERINGS AND GROUP THEORY
- CHAPTER 8 BASS-SERRE THEORY
- CHAPTER 9 DECISION PROBLEMS
- CHAPTER 10 FURTHER TOPICS
- NOTES AND REFERENCES
- BIBLIOGRAPHY
- INDEX
Summary
SOME POINT-SET TOPOLOGY
We begin with a lemma that will be frequently used, often without explicit mention.
Lemma 1 (Glueing Lemma) (i) Let X and Y be sets, let Xαbe subsets of X such that X-∪Xα, and let fα:Xα → Y be functions such that fα|Xα∩Xβ – fβ|Xα∩Xβfor all α and β. Then there is a unique function f:X→Y such that f|Xα – fαfor all α.
(ii) Let the conditions of (i) hold, and let X and Y be topological spaces. Suppose that each fαis continuous (when Xαis given the subspace topology). Suppose either that there are only finitely many sets Xαeach of which is a closed subspace of X or that each Xαis an open subspace of X. Then f is continuous.
Remark When f is a function from a set x to a set Y and A is a subset of X the notation f\A means the restriction of f to A; that is, the function from A to Y whose value on a ∈ A is fa.
Proof (i) Let S⊆XxY be {(x,y); there is α such that x∈Xα and y-xfα). Since X-⊃Xα, for every x there is at least one y with (x,y)∈S. Suppose that (x,y)∈S and (x,y)∈S. Then there are α and β with x∈Xα, y - xfα, and x∈Xβ, z - xfβ. Since fsub>α - fβ on fsub>α ∩ fβ, by hypothesis, it follows that y - z.
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- Information
- Combinatorial Group TheoryA Topological Approach, pp. 49 - 61Publisher: Cambridge University PressPrint publication year: 1989