General topology | Homotopy theory
In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space X is compactly generated if it satisfies the following condition: A subspace A is closed in X if and only if A ∩ K is closed in K for all compact subspaces K ⊆ X. Equivalently, one can replace closed with open in this definition. If X is coherent with any cover of compact subspaces in the above sense then it is, in fact, coherent with all compact subspaces. A Hausdorff-compactly generated space or k-space is a topological space whose topology is coherent with the family of all compact Hausdorff subspaces. Sometimes in the literature a compactly generated space refers to a Hausdorff-compactly generated space. In these cases compactness is often explicitly redefined at the beginning to mean both compact and Hausdorff (and quasi-compact takes the meaning of compact). In this article we make a clear separation between compactly generated spaces and Hausdorff-compactly generated spaces, since the choice affects the statement of the associated theorems. A compactly generated Hausdorff space is a compactly generated space that is also Hausdorff. This is not to be confused with a Hausdorff-compactly generated space which may or may not be Hausdorff. Like many compactness conditions, compactly generated spaces are often assumed to be Hausdorff or weakly Hausdorff. See the category of compactly generated weak Hausdorff spaces for the use in algebraic topology. (Wikipedia).
Math 101 Fall 2017 112917 Introduction to Compact Sets
Definition of an open cover. Definition of a compact set (in the real numbers). Examples and non-examples. Properties of compact sets: compact sets are bounded. Compact sets are closed. Closed subsets of compact sets are compact. Infinite subsets of compact sets have accumulation poi
From playlist Course 6: Introduction to Analysis (Fall 2017)
What exactly is space? Brian Greene explains what the "stuff" around us is. Subscribe to our YouTube Channel for all the latest from World Science U. Visit our Website: http://www.worldscienceu.com/ Like us on Facebook: https://www.facebook.com/worldscienceu Follow us on Twitter: https:
From playlist Science Unplugged: Physics
Every Compact Set in n space is Bounded
Every Compact Set in n space is Bounded If you enjoyed this video please consider liking, sharing, and subscribing. You can also help support my channel by becoming a member https://www.youtube.com/channel/UCr7lmzIk63PZnBw3bezl-Mg/join Thank you:)
From playlist Advanced Calculus
Compact sets enjoy some mysterious properties, which I'll discuss in this video. More precisely, compact sets are always bounded and closed. The beauty of this result lies in the proof, which is an elegant application of this subtle concept. Enjoy! Compactness Definition: https://youtu.be
From playlist Topology
A Dyson Sphere is a megastructure that could be built around a star to harness all the solar energy it gives off. In this video we talk about the different kinds of Dyson Spheres, Dyson Clouds and other megastructures that could be built - and how we might even detect them from Earth. ht
From playlist Guide to Space
Math 101 Introduction to Analysis 112515: Introduction to Compact Sets
Introduction to Compact Sets: open covers; examples of finite and infinite open covers; definition of compactness; example of a non-compact set; compact implies closed; closed subset of compact set is compact; continuous image of a compact set is compact
From playlist Course 6: Introduction to Analysis
Math 131 092116 Properties of Compact Sets
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From playlist Course 7: (Rudin's) Principles of Mathematical Analysis
Every Closed Subset of a Compact Space is Compact Proof
Every Closed Subset of a Compact Space is Compact Proof If you enjoyed this video please consider liking, sharing, and subscribing. You can also help support my channel by becoming a member https://www.youtube.com/channel/UCr7lmzIk63PZnBw3bezl-Mg/join Thank you:)
From playlist Topology
Mather-Thurston’s theory, non abelian Poincare duality and diffeomorphism groups - Sam Nariman
Workshop on the h-principle and beyond Topic: Mather-Thurston’s theory, non abelian Poincare duality and diffeomorphism groups Speaker: Sam Nariman Affiliation: Purdue University Date: November 1, 2021 Abstract: I will discuss a remarkable generalization of Mather’s theorem by Thurston
From playlist Mathematics
Homological generalizations of trace - Dmitry Vaintrob
Topic: Homological generalizations of trace Speaker: Dmitry Vaintrob, Member, School of Mathematics Time/Room: 4:15pm - 4:30pm/S-101 More videos on http://video.ias.edu
From playlist Mathematics
Tropical motivic integration - S. Payne - Workshop 2 - CEB T1 2018
Sam Payne (Yale University) / 09.03.2018 Tropical motivic integration. I will present a new tool for the calculation of motivic invariants appearing in Donaldson-Thomas theory, such as the motivic Milnor fiber and motivic nearby fiber, starting from a theory of volumes of semi-algebraic
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
Cornelia Drutu - Connections between hyperbolic geometry and median geometry
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From playlist Geometry in non-positive curvature and Kähler groups
Ana Caraiani: Vanishing theorems for Shimura varieties at infinite level
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From playlist Algebraic and Complex Geometry
Geordie Williamson: Geometric Representation Theory and the Geometric Satake Equivalence
MSI Virtual Colloquium: Geometric Representation Theory and the Geometric Satake Equivalence Geordie Williamson (University of Sydney) During this colloquium Geordie will explain in very broad terms, what the Langlands correspondence is and why people care about it. He will then explain i
From playlist Geordie Williamson: Representation theory and the Geometric Satake
Math 101 Introduction to Analysis 113015: Compact Sets, ct'd
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From playlist Course 6: Introduction to Analysis
A New Cubulation Theorem for Hyperbolic Groups- Daniel Groves
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From playlist Geometric Structures on 3-manifolds
Colloquium MathAlp 2019 - Claude Lebrun
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From playlist Colloquiums MathAlp
Generalized Radon transforms in tomography
Advanced Instructional School on Theoretical and Numerical Aspects of Inverse Problems URL: https://www.icts.res.in/program/IP2014 Dates: Monday 16 Jun, 2014 - Saturday 28 Jun, 2014 Description In Inverse Problems the goal is to determine the properties of the interior of an object from
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Floer Cohomology and Arc Spaces - Mark McLean
IAS/PU-Montreal-Paris-Tel-Aviv Symplectic Geometry Topic: Floer Cohomology and Arc Spaces Speaker: Mark McLean Affiliation: Stony Brook University Date: June 12, 2020 For more video please visit http://video.ias.edu
From playlist Mathematics
Is there any place in the Universe where there's truly nothing? Consider the gaps between stars and galaxies? Or the gaps between atoms? What are the properties of nothing?
From playlist Guide to Space