True of False. 1. The order topology of a simply ordered set X is the standard topology defined using the order relation. 2. Both open and closed rays of an ordered set X form a subbasis for the order topol- ogy. 3. Every susbsapce of a discrete space is also discrete.
True of False. 1. The order topology of a simply ordered set X is the standard topology defined using the order relation. 2. Both open and closed rays of an ordered set X form a subbasis for the order topol- ogy. 3. Every susbsapce of a discrete space is also discrete.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:. True of False.
1. The order topology of a simply ordered set X is the standard topology defined using
the order relation.
2. Both open and closed rays of an ordered set X form a subbasis for the order topol-
ogy.
3. Every susbsapce of a discrete space is also discrete.
4. Every subspace of an indiscrete is also indiscrete.
5. Given a box topology on B = Xa, the product topology is always finer than the
box topology.
6. The closure A of a subset A of a metric space X with metric d is the set of points
Ā = {r\d(x, A) > }.
7. Let a function d be a metric on a set X. If a, b, c € X are all distinct, then
d*(a, c) < d*(a, b6) + d*(b, c).
8. The subset A = {(x, y)|x² – y 2 4} of R is connected.
9. A connected space X has only one component.
10. Every indiscrete space is connected.
11. There exist a space containing only finite elements which is not compact.
12. If E is compact and F is closed, then EnF is compact.
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