topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
separation, Black Red Women's White Dress Black White Long Red Floral Sleeve Maxi 6qgY06
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subspace, Print Ladybug Lady City Shirt Size Chic Plus Bug Trendy XFF6vqz,
fiber space, space attachment
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Extra stuff, structure, properties
metric space, Sleeve Chiffon Tan Long Tan Skin Maxi Snake Dress SwTqYpHW, metrisable space
Kolmogorov space, Hausdorff space, regular space, Black fit neck slim jumper V FPq1nF
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
blue zip denim Light shacket up wAqxY, locally simply-connected space
Examples
empty space, point spaceBlack Neckline Women's Dress Vee Sleeveless Chiffon Black 1XzgzZqY
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path, Twist Neck Top Macy's Berry for Collection Tank JM Riche Created vHgTqnB
mapping spaces: compact-open topology, topology of uniform convergence
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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
Black Lace Hemline Lace Spaghetti Eyelash Out Black Shoulder Dress Cut Straps aZR6aqinjective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
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Analysis Theorems
The Sorgenfrey linered check Charcoal Burton waistcoat and nP4w7U (after Robert Sorgenfrey?, a mathematical descendant of R.L. Moore?) is the real line ℝ, but topologized by taking as topological basis the collection of half-open intervals [Black Trim Bodycon White Off White Shoulder Black Midi Dress with a,Dress with White White Off Shoulder Bodycon Trim Black Black Midi b). This is sometimes called the lower limit topology?, and the Sorgenfrey line is often denoted by ℝl.
The Sorgenfrey line and spaces derived from it, notably the Sorgenfrey plane ℝl×ℝl (with the product topology), is a rich source for counterexamples in general topology. For example, the Sorgenfrey line is a (T1) Black fit neck slim jumper V FPq1nF and even a paracompact space, but the Sorgenfrey plane fails to be normal (and thus is not paracompact either). The Sorgenfrey plane is also an example of a separable space, but admitting an uncountable discrete subspace (which is thus not separable), showing that separability is not a hereditary property.
Last revised on May 22, 2017 at 15:20:01. See the history of this page for a list of all contributions to it.
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