Hausdorff Example 3: Function Spaces
Point Set Topology: For a third example, we consider function spaces. We begin with the space of continuous functions on [0,1]. As a metric space, this example is Hausdorff, but not complete. We consider Cauchy sequences and a possible completion.
From playlist Point Set Topology
Math 131 092816 Continuity; Continuity and Compactness
Review definition of limit. Definition of continuity at a point; remark about isolated points; connection with limits. Composition of continuous functions. Alternate characterization of continuous functions (topological definition). Continuity and compactness: continuous image of a com
From playlist Course 7: (Rudin's) Principles of Mathematical Analysis
Math 131 Fall 2018 101018 Continuity and Compactness
Definition: bounded function. Continuous image of compact set is compact. Continuous image in Euclidean space of compact set is bounded. Extreme Value Theorem. Continuous bijection on compact set has continuous inverse. Definition of uniform continuity. Continuous on compact set impl
From playlist Course 7: (Rudin's) Principles of Mathematical Analysis (Fall 2018)
Introduction to Discrete and Continuous Functions
This video defines and provides examples of discrete and continuous functions.
From playlist Introduction to Functions: Function Basics
Topology Proof The Constant Function is Continuous
Topology Proof The Constant Function is Continuous 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
Calculus - Continuous functions
This video will describe how calculus defines a continuous function using limits. Some examples are used to find where a function is continuous, and where it is not continuous. Remember to check that the value at c and the limit as x approaches c exist, and agree. For more videos please
From playlist Calculus
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
Determine the Intervals on which the Function is Continuous Four Examples
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Determine the Intervals on which the Function is Continuous Four Examples
From playlist Calculus 1 Exam 1 Playlist
Determine Where the Function is Not Continuous
In this video I will show you how to Determine Where the Function is Not Continuous.
From playlist Continuity Problems
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
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
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
MAST30026 Lecture 11: Hausdorff spaces (Part 1)
I introduced the Hausdorff condition, proved some basic properties, discussed the "real line with a double point" as an example of a non-Hausdorff space, proved that a compact subspace of a Hausdorff space is closed, and that continuous bijections from compact to Hausdorff spaces are homeo
From playlist MAST30026 Metric and Hilbert spaces
MAST30026 Lecture 17: Integrals
I began by explaining how, in order to work with infinite-dimensional function spaces constructively, we need to use integrals. Then I defined an "integral pair", showed that the Riemann integral on a closed interval is an example, and proved that there is a product operation on integral p
From playlist MAST30026 Metric and Hilbert spaces
MAST30026 Lecture 18: Banach spaces (Part 1)
There are many Lipschitz equivalent metrics on Euclidean space, apart from the sup-metric (which we have successfully generalised to function spaces) there are also metrics defined using sums. To generalise those, we need integrals, and the resulting theory leads to Banach spaces. In this
From playlist MAST30026 Metric and Hilbert spaces
Geometry of Surfaces - Topological Surfaces Lecture 1 : Oxford Mathematics 3rd Year Student Lecture
This is the first of four lectures from Dominic Joyce's 3rd Year Geometry of Surfaces course. The four lectures cover topological surfaces and conclude with a big result, namely the classification of surfaces. This lecture provides an introduction to the course and to topological surfaces.
From playlist Oxford Mathematics Student Lectures - Geometry of Surfaces
MAST30026 Lecture 16: Stone-Weierstrass theorem (Part 2)
In this lecture I introduced the algebra structure on spaces of real-valued functions, and proved the Stone-Weierstrass theorem about dense subalgebras of this algebra. Lecture notes: http://therisingsea.org/notes/mast30026/lecture16.pdf The class webpage: http://therisingsea.org/post/mas
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?
Math 131 Fall 2018 100818 Limits and Continuity in Metric Spaces
Limits of functions (in the setting of metric spaces). Definition. Rephrasal of definition. Uniqueness of limit. Definition of continuity at a point. Remark on continuity at an isolated point. Relation with limits. Composition of continuous functions is continuous. Alternate (topol
From playlist Course 7: (Rudin's) Principles of Mathematical Analysis (Fall 2018)