order topology in nLab
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Context
Topology
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
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fiber space, space attachment
Extra stuff, structure, properties
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Kolmogorov space, Hausdorff space, regular space, normal space
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sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
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closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
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open subspaces of compact Hausdorff spaces are locally compact
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compact spaces equivalently have converging subnet of every net
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continuous metric space valued function on compact metric space is uniformly continuous
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paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
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injective proper maps to locally compact spaces are equivalently the closed embeddings
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locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
Contents
Definition
Given a linearly ordered set , its order topology is the topology on generated from the sub-basis
whose elements are the “open half rays”
Examples
- The usual Euclidean space metric topology on the real numbers is equal to the order topology with respect to the canonical ordering of the real numbers. The analogous statement then holds for the rational numbers equipped with their subspace topology.
References
- Wikipedia, Order topology
Last revised on May 3, 2017 at 21:44:21. See the history of this page for a list of all contributions to it.