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3. Show that every locally compact space is compactly generated.

3. Show that every locally compact space is compactly generated.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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**Problem 3:** Show that every locally compact space is compactly generated. *Explanation:* This statement is a topic in topology, a branch of mathematics concerned with the properties of space that are preserved under continuous transformations. A "locally compact space" is a space in which every point has a neighborhood that is compact, meaning it is both closed and bounded within the space. A "compactly generated space" is a space where the topology can be determined by compact subsets. To prove this, one typically demonstrates that the topology of the space can indeed be constructed from its compact subsets. This often involves using the fact that the union of compact sets retains certain desirable properties, making the space easier to manage and understand in a topological sense. *Note:* There are no graphs or diagrams associated with this problem, as it primarily involves theoretical concepts and proofs.
Transcribed Image Text:**Problem 3:** Show that every locally compact space is compactly generated. *Explanation:* This statement is a topic in topology, a branch of mathematics concerned with the properties of space that are preserved under continuous transformations. A "locally compact space" is a space in which every point has a neighborhood that is compact, meaning it is both closed and bounded within the space. A "compactly generated space" is a space where the topology can be determined by compact subsets. To prove this, one typically demonstrates that the topology of the space can indeed be constructed from its compact subsets. This often involves using the fact that the union of compact sets retains certain desirable properties, making the space easier to manage and understand in a topological sense. *Note:* There are no graphs or diagrams associated with this problem, as it primarily involves theoretical concepts and proofs.
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