topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
Compact ordered spaces can be thought of as a refinement of the notion of compact Hausdorff space to ordered spaces. They share many properties of compact Hausdorff spaces, where equality is replaced by the order relation. These spaces are a convenient choice for notions of convergence in an ordered setting.
In particular, just as compact Hausdorff spaces can be seen as algebras of the ultrafilter monad on Set, compact ordered spaces can be seen as algebras of the prime upper filter monad? on Pos, (see below).
The concept was introduced by Nachbin (see Nachbin ‘65).
A compact ordered space or compact pospace is a compact topological space $X$ equipped with a partial order which is closed as a subset of $X\times X$.
A compact ordered space is always Hausdorff, i.e. it is a compactum. To see this, note that since the relation is a closed subset of $X\times X$, so is the opposite relation, and hence their intersection too, which is the diagonal of $X\times X$.
Conversely, a compact Hausdorff space can be seen as a compact ordered space with the discrete order?.
In a compact ordered space, the up-sets and down-sets $\uparrow\!\{x\}$ and $\downarrow\!\{x\}$ are always closed and compact.
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A canonical choice of morphisms between compact ordered spaces is continuous, monotone maps, which form a category. This category is usually just called the category of compact ordered spaces, and denoted by $CompOrd$.
With the 2-cell structure given by the pointwise order, $CompOrd$ becomes a locally posetal 2-category.
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For now, see Flagg ‘96.
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For now, see Jung ‘04.
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Leopoldo Nachbin, Topology and Order, Van Nostrand, 1965.
Bob Flagg, Algebraic theories of compact pospaces, Topology and its Applications, 1996 (doi:10.1016/S0166-8641(96)00117-4).
Achim Jung, Stably compact spaces and the probabilistic powerspace construction, ENTCS 87, 2004 (doi:10.1016/j.entcs.2004.10.001).
Last revised on October 25, 2019 at 15:25:35. See the history of this page for a list of all contributions to it.