Algebraic Topology - 5.1 - Mappings Spaces and the Compact Open Topology
We define the compact open topology on mapping spaces.
From playlist Algebraic Topology
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
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
This video is about compactness and some of its basic properties.
From playlist Basics: Topology
I define closed sets, an important notion in topology and analysis. It is defined in terms of limit points, and has a priori nothing to do with open sets. Yet I show the important result that a set is closed if and only if its complement is open. More topology videos can be found on my pla
From playlist Topology
Algebraic Topology - 5.3 - Mapping Spaces and the Compact Open Topology
Description of the adjunction (X \times -, Top(X,-))
From playlist Algebraic Topology
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)
Topology 1.1 : Open Sets of Reals
In this video, I give a definition of the open sets on the real numbers. Email : fematikaqna@gmail.com Code : https://github.com/Fematika/Animations Notes : None yet
From playlist Topology
Here I give a taste of topology by defining the notion of an open set, give examples, and show its main properties. I further define the notion of an interior. Enjoy this little topology and analysis extravaganza. More videos can be found on my playlist below. Closed Sets: https://youtu.b
From playlist Topology
MAST30026 Lecture 12: Function spaces (Part 3)
We continued the discussion of the compact-open topology on function spaces. Guided by Part 2 we defined this topology, and got about half way through the proof that the adjunction property (aka the exponential law) holds when function spaces are given this topology. Lecture notes: http:/
From playlist MAST30026 Metric and Hilbert spaces
What is a Manifold? Lesson 15: The cylinder as a quotient space
What is a Manifold? Lesson 15: The cylinder as a quotient space This lesson covers several different ideas on the way to showing how the cylinder can be described as a quotient space. Lot's of ideas in this lecture! ... too many probably....
From playlist What is a Manifold?
MAST30026 Lecture 12: Function spaces (Part 4)
We completed the proof that the adjunction property holds for the space of continuous functions from a locally compact Hausdorff space, reminded ourselves of some of the immediate consequences of this theorem, and then began motivating the construction of a metric on function spaces. Lect
From playlist MAST30026 Metric and Hilbert spaces
What is a Manifold? Lesson 6: Topological Manifolds
Topological manifolds! Finally! I had two false starts with this lesson, but now it is fine, I think.
From playlist What is a Manifold?
What is a Manifold? Lesson 5: Compactness, Connectedness, and Topological Properties
The last lesson covering the topological prep-work required before we begin the discussion of manifolds. Topics covered: compactness, connectedness, and the relationship between homeomorphisms and topological properties.
From playlist What is a Manifold?
MAST30026 Lecture 13: Metrics on function spaces (Part 1)
I defined the sup metric on the function space Cts(X,Y) where X is compact and Y is a metric space, and proved that the associated metric topology agrees with the compact-open topology. Lecture notes: http://therisingsea.org/notes/mast30026/lecture13.pdf The class webpage: http://therisi
From playlist MAST30026 Metric and Hilbert spaces
MAST30026 Lecture 12: Function spaces (Part 2)
The aim of this lecture was to motivate the definition of the compact-open topology on function spaces, via the adjunction property. I explained how any topology making the adjunction property true must include a certain class of open sets, which we will define next lecture to be a sub-bas
From playlist MAST30026 Metric and Hilbert spaces
Algebraic Topology - 1 - Compact Hausdorff Spaces (a Review of Point-Set Topology)
This is mostly a review point set topology. In general it is not true that a bijective continuous map is invertible (you need to worry about the inverse being continuous). In the case that your spaces are compact hausdorff this is true! We prove this in this video and review necessary fac
From playlist Algebraic Topology
Topology PhD Qualifying Exam Problems (Stream 1)
Just practicing some arguments from topology qualifying exam problems. A few folks said they wanted me to hang out here instead of on Twitch today. 00:00:00 Dead Air 00:00:53 I exist huzzah! 00:09:26 Continuous Images of Metric Spaces in Hausdorff Spaces Problem 01:13:45 Separable First C
From playlist CHALK Streams
Algebraic Topology - 5.2 - Mapping Spaces and the Compact Open Topology
We give an example of how the metric topology agrees with the compact open topology on Top(X,Y) when X is compact and Y is metric.
From playlist Algebraic Topology
Manifolds - Part 8 - Compactness
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From playlist Manifolds