N.B. Some authors use the name Top for the category with topological manifolds as objects and continuous maps as morphisms.
As a concrete category
Like many categories, the category Top is a concrete category (also known as a construct), meaning its objects are sets with additional structure (i.e. topologies) and its morphisms are functions preserving this structure. There is a natural forgetful functor
to the category of sets which assigns to each topological space the underlying set and to each continuous map the underlying function.
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, that satisfy a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.
Definition
The utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure. Thus one chooses the axiomatisation suited for the application. The most commonly used, and the most elegant, is that in terms of open sets, but the most intuitive is that in terms of neighbourhoods and so we give this first.
Note: A variety of other axiomatisations of topological spaces are listed in the Exercises of the book by Vaidyanathaswamy.
Introductory video on topology that explains the central role of topological spaces in mathematics. Examples include indiscrete and discrete topologies on a finite set X = {1,2,3}
Recommended resource for topology:
* Elementary Topology Problem Textbook (O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, V. M. Kharlamov)
Link: http://www.pdmi.ras.ru/~olegviro/topoman/eng-book-nopfs.pdf
published: 12 Apr 2020
Definition of Topology and topological space with example
Best definition of topology and topological space with examples. important notes on general topology.
If you like this video and you want us to make more videos. A little dollar will go a long way in supporting our work.
Click on the link below to make donations.
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published: 15 Feb 2020
Topology Lecture 01: Topological Spaces
We define topological spaces and give examples including the discrete, trivial, and metric topologies.
00:00 Introduction
00:39 Reference and Prerequisites
02:17 Motivation: Familiar Spaces
10:22 Definition: Topological Space
19:59 Example: Discrete Topology
23:56 Example: Trivial Topology
27:51 Example: A Small Topology
32:41 Example: Metric Topology
37:21 Common Euclidean Subspaces
This lecture follows Lee's "Introduction to Topological Manifolds", chapter 2.
A playlist with all the videos in this series can be found here:
https://www.youtube.com/playlist?list=PLd8NbPjkXPliJunBhtDNMuFsnZPeHpm-0
published: 15 Feb 2021
Definition of a Topological Space
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys
Definition of a Topological Space
published: 30 Nov 2015
Weird Topological Spaces // Connected vs Path Connected vs Simply Connected
Keep learning at ► https://brilliant.org/TreforBazett. Get started for free for 30 days — and the first 200 people get 20% off an annual premium subscription!
What exactly does it mean for a space to be connected? In this video we will contrast the notions of connected, path connected and simply connected for a range of topological spaces such as the topologists sine curve, the torus, and the fascinating Alexander's Horned Sphere.
0:00 Topologist's Sine Curve
1:25 Definition of Connected
3:10 Definition of Path Connected
4:06 Topologist's Sine Curve again
7:24 Simple Connected
9:37 Alexander's Horned Sphere
11:55 Brilliant.org/TreforBazett
The animation of Alexander's Horned Sphere was created by Gian Todesco: https://en.wikipedia.org/wiki/Alexander_horned_sphere#/media/File:Construcci...
I will cover the topology of the real line and the definition of continuous.
published: 28 Oct 2012
What is Hausdorff Topological Space?
In this video, we are going to define Hausdorff topological space and discuss a couple of examples.
If you like the video, please help my channel grow by subscribing to my channel and sharing my videos.
Thank you for watching!
Maksym Zubkov
[email protected]
-~-~~-~~~-~~-~-
Please watch: "Real Projective Space, n=1"
https://www.youtube.com/watch?v=2ottRuDA5WA
-~-~~-~~~-~~-~-
Twitter: https://www.twitter.com/maksym_zubkov
Instagram: https://www.instagram.com/m.zubkov_
Math Blog: https://maksymzubkov.com
published: 23 Jun 2021
Topological Spaces Part 1
In this video we motivate and define the concept of a topological space.
published: 26 Jul 2015
A topological proof of the infinitude of the primes
In this video I summarize Furstenberg's topological proof of the infinitude of the prime numbers. #math #stem #proof #numbertheory #primenumber #topology
published: 19 Oct 2024
Examples of topological spaces (Lesson 2)
This is a second video on the study of Topological Spaces.
Here, we try to learn how to determine whether a collection of subsets is a topology on X or not.
You can find the link to the previous video here;
https://www.youtube.com/watch?v=snMGzSffMDM
Introductory video on topology that explains the central role of topological spaces in mathematics. Examples include indiscrete and discrete topologies on a fin...
Introductory video on topology that explains the central role of topological spaces in mathematics. Examples include indiscrete and discrete topologies on a finite set X = {1,2,3}
Recommended resource for topology:
* Elementary Topology Problem Textbook (O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, V. M. Kharlamov)
Link: http://www.pdmi.ras.ru/~olegviro/topoman/eng-book-nopfs.pdf
Introductory video on topology that explains the central role of topological spaces in mathematics. Examples include indiscrete and discrete topologies on a finite set X = {1,2,3}
Recommended resource for topology:
* Elementary Topology Problem Textbook (O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, V. M. Kharlamov)
Link: http://www.pdmi.ras.ru/~olegviro/topoman/eng-book-nopfs.pdf
Best definition of topology and topological space with examples. important notes on general topology.
If you like this video and you want us to make more video...
Best definition of topology and topological space with examples. important notes on general topology.
If you like this video and you want us to make more videos. A little dollar will go a long way in supporting our work.
Click on the link below to make donations.
https://www.youtube.com/redirect?q=https%3A%2F%2Fwww.paypal.com%2Fcgi-bin%2Fwebscr%3Fcmd%3D_s-xclick%26hosted_button_id%3DFU38QVXYMU452%26source%3Durl&event=channel_banner&redir_token=6wc1nnmmVXbZZ6Xql1US8hIeJ3N8MTU4NzgyMjIwNEAxNTg3NzM1ODA0
Best definition of topology and topological space with examples. important notes on general topology.
If you like this video and you want us to make more videos. A little dollar will go a long way in supporting our work.
Click on the link below to make donations.
https://www.youtube.com/redirect?q=https%3A%2F%2Fwww.paypal.com%2Fcgi-bin%2Fwebscr%3Fcmd%3D_s-xclick%26hosted_button_id%3DFU38QVXYMU452%26source%3Durl&event=channel_banner&redir_token=6wc1nnmmVXbZZ6Xql1US8hIeJ3N8MTU4NzgyMjIwNEAxNTg3NzM1ODA0
We define topological spaces and give examples including the discrete, trivial, and metric topologies.
00:00 Introduction
00:39 Reference and Prerequisites
02:...
We define topological spaces and give examples including the discrete, trivial, and metric topologies.
00:00 Introduction
00:39 Reference and Prerequisites
02:17 Motivation: Familiar Spaces
10:22 Definition: Topological Space
19:59 Example: Discrete Topology
23:56 Example: Trivial Topology
27:51 Example: A Small Topology
32:41 Example: Metric Topology
37:21 Common Euclidean Subspaces
This lecture follows Lee's "Introduction to Topological Manifolds", chapter 2.
A playlist with all the videos in this series can be found here:
https://www.youtube.com/playlist?list=PLd8NbPjkXPliJunBhtDNMuFsnZPeHpm-0
We define topological spaces and give examples including the discrete, trivial, and metric topologies.
00:00 Introduction
00:39 Reference and Prerequisites
02:17 Motivation: Familiar Spaces
10:22 Definition: Topological Space
19:59 Example: Discrete Topology
23:56 Example: Trivial Topology
27:51 Example: A Small Topology
32:41 Example: Metric Topology
37:21 Common Euclidean Subspaces
This lecture follows Lee's "Introduction to Topological Manifolds", chapter 2.
A playlist with all the videos in this series can be found here:
https://www.youtube.com/playlist?list=PLd8NbPjkXPliJunBhtDNMuFsnZPeHpm-0
Keep learning at ► https://brilliant.org/TreforBazett. Get started for free for 30 days — and the first 200 people get 20% off an annual premium subscription!
...
Keep learning at ► https://brilliant.org/TreforBazett. Get started for free for 30 days — and the first 200 people get 20% off an annual premium subscription!
What exactly does it mean for a space to be connected? In this video we will contrast the notions of connected, path connected and simply connected for a range of topological spaces such as the topologists sine curve, the torus, and the fascinating Alexander's Horned Sphere.
0:00 Topologist's Sine Curve
1:25 Definition of Connected
3:10 Definition of Path Connected
4:06 Topologist's Sine Curve again
7:24 Simple Connected
9:37 Alexander's Horned Sphere
11:55 Brilliant.org/TreforBazett
The animation of Alexander's Horned Sphere was created by Gian Todesco: https://en.wikipedia.org/wiki/Alexander_horned_sphere#/media/File:Construcci%C3%B3n_de_esfera_de_Alexander_con_cuernos.gif
Check out my MATH MERCH line in collaboration with Beautiful Equations
►https://beautifulequations.net/pages/trefor
COURSE PLAYLISTS:
►DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS
►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6
►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m
► CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n
►MULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd
►VECTOR CALCULUS (Calc IV) https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa
►DIFFERENTIAL EQUATIONS: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxde-SlgmWlCmNHroIWtujBw
►LAPLACE TRANSFORM: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1
►GAME THEORY: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxdzD8KpTHz6_gsw9pPxRFlX
OTHER PLAYLISTS:
► Learning Math Series
https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw
►Cool Math Series:
https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho
BECOME A MEMBER:
►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join
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SOCIALS:
►Twitter (math based): http://twitter.com/treforbazett
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Keep learning at ► https://brilliant.org/TreforBazett. Get started for free for 30 days — and the first 200 people get 20% off an annual premium subscription!
What exactly does it mean for a space to be connected? In this video we will contrast the notions of connected, path connected and simply connected for a range of topological spaces such as the topologists sine curve, the torus, and the fascinating Alexander's Horned Sphere.
0:00 Topologist's Sine Curve
1:25 Definition of Connected
3:10 Definition of Path Connected
4:06 Topologist's Sine Curve again
7:24 Simple Connected
9:37 Alexander's Horned Sphere
11:55 Brilliant.org/TreforBazett
The animation of Alexander's Horned Sphere was created by Gian Todesco: https://en.wikipedia.org/wiki/Alexander_horned_sphere#/media/File:Construcci%C3%B3n_de_esfera_de_Alexander_con_cuernos.gif
Check out my MATH MERCH line in collaboration with Beautiful Equations
►https://beautifulequations.net/pages/trefor
COURSE PLAYLISTS:
►DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS
►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6
►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m
► CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n
►MULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd
►VECTOR CALCULUS (Calc IV) https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa
►DIFFERENTIAL EQUATIONS: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxde-SlgmWlCmNHroIWtujBw
►LAPLACE TRANSFORM: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1
►GAME THEORY: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxdzD8KpTHz6_gsw9pPxRFlX
OTHER PLAYLISTS:
► Learning Math Series
https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw
►Cool Math Series:
https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho
BECOME A MEMBER:
►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join
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SOCIALS:
►Twitter (math based): http://twitter.com/treforbazett
►Instagram (photography based): http://instagram.com/treforphotography
In this video, we are going to define Hausdorff topological space and discuss a couple of examples.
If you like the video, please help my channel grow by subsc...
In this video, we are going to define Hausdorff topological space and discuss a couple of examples.
If you like the video, please help my channel grow by subscribing to my channel and sharing my videos.
Thank you for watching!
Maksym Zubkov
[email protected]
-~-~~-~~~-~~-~-
Please watch: "Real Projective Space, n=1"
https://www.youtube.com/watch?v=2ottRuDA5WA
-~-~~-~~~-~~-~-
Twitter: https://www.twitter.com/maksym_zubkov
Instagram: https://www.instagram.com/m.zubkov_
Math Blog: https://maksymzubkov.com
In this video, we are going to define Hausdorff topological space and discuss a couple of examples.
If you like the video, please help my channel grow by subscribing to my channel and sharing my videos.
Thank you for watching!
Maksym Zubkov
[email protected]
-~-~~-~~~-~~-~-
Please watch: "Real Projective Space, n=1"
https://www.youtube.com/watch?v=2ottRuDA5WA
-~-~~-~~~-~~-~-
Twitter: https://www.twitter.com/maksym_zubkov
Instagram: https://www.instagram.com/m.zubkov_
Math Blog: https://maksymzubkov.com
In this video I summarize Furstenberg's topological proof of the infinitude of the prime numbers. #math #stem #proof #numbertheory #primenumber #topology
In this video I summarize Furstenberg's topological proof of the infinitude of the prime numbers. #math #stem #proof #numbertheory #primenumber #topology
In this video I summarize Furstenberg's topological proof of the infinitude of the prime numbers. #math #stem #proof #numbertheory #primenumber #topology
This is a second video on the study of Topological Spaces.
Here, we try to learn how to determine whether a collection of subsets is a topology on X or not.
Yo...
This is a second video on the study of Topological Spaces.
Here, we try to learn how to determine whether a collection of subsets is a topology on X or not.
You can find the link to the previous video here;
https://www.youtube.com/watch?v=snMGzSffMDM
This is a second video on the study of Topological Spaces.
Here, we try to learn how to determine whether a collection of subsets is a topology on X or not.
You can find the link to the previous video here;
https://www.youtube.com/watch?v=snMGzSffMDM
Introductory video on topology that explains the central role of topological spaces in mathematics. Examples include indiscrete and discrete topologies on a finite set X = {1,2,3}
Recommended resource for topology:
* Elementary Topology Problem Textbook (O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, V. M. Kharlamov)
Link: http://www.pdmi.ras.ru/~olegviro/topoman/eng-book-nopfs.pdf
Best definition of topology and topological space with examples. important notes on general topology.
If you like this video and you want us to make more videos. A little dollar will go a long way in supporting our work.
Click on the link below to make donations.
https://www.youtube.com/redirect?q=https%3A%2F%2Fwww.paypal.com%2Fcgi-bin%2Fwebscr%3Fcmd%3D_s-xclick%26hosted_button_id%3DFU38QVXYMU452%26source%3Durl&event=channel_banner&redir_token=6wc1nnmmVXbZZ6Xql1US8hIeJ3N8MTU4NzgyMjIwNEAxNTg3NzM1ODA0
We define topological spaces and give examples including the discrete, trivial, and metric topologies.
00:00 Introduction
00:39 Reference and Prerequisites
02:17 Motivation: Familiar Spaces
10:22 Definition: Topological Space
19:59 Example: Discrete Topology
23:56 Example: Trivial Topology
27:51 Example: A Small Topology
32:41 Example: Metric Topology
37:21 Common Euclidean Subspaces
This lecture follows Lee's "Introduction to Topological Manifolds", chapter 2.
A playlist with all the videos in this series can be found here:
https://www.youtube.com/playlist?list=PLd8NbPjkXPliJunBhtDNMuFsnZPeHpm-0
Keep learning at ► https://brilliant.org/TreforBazett. Get started for free for 30 days — and the first 200 people get 20% off an annual premium subscription!
What exactly does it mean for a space to be connected? In this video we will contrast the notions of connected, path connected and simply connected for a range of topological spaces such as the topologists sine curve, the torus, and the fascinating Alexander's Horned Sphere.
0:00 Topologist's Sine Curve
1:25 Definition of Connected
3:10 Definition of Path Connected
4:06 Topologist's Sine Curve again
7:24 Simple Connected
9:37 Alexander's Horned Sphere
11:55 Brilliant.org/TreforBazett
The animation of Alexander's Horned Sphere was created by Gian Todesco: https://en.wikipedia.org/wiki/Alexander_horned_sphere#/media/File:Construcci%C3%B3n_de_esfera_de_Alexander_con_cuernos.gif
Check out my MATH MERCH line in collaboration with Beautiful Equations
►https://beautifulequations.net/pages/trefor
COURSE PLAYLISTS:
►DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS
►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6
►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m
► CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n
►MULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd
►VECTOR CALCULUS (Calc IV) https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa
►DIFFERENTIAL EQUATIONS: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxde-SlgmWlCmNHroIWtujBw
►LAPLACE TRANSFORM: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1
►GAME THEORY: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxdzD8KpTHz6_gsw9pPxRFlX
OTHER PLAYLISTS:
► Learning Math Series
https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw
►Cool Math Series:
https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho
BECOME A MEMBER:
►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join
MATH BOOKS I LOVE (affilliate link):
► https://www.amazon.com/shop/treforbazett
SOCIALS:
►Twitter (math based): http://twitter.com/treforbazett
►Instagram (photography based): http://instagram.com/treforphotography
In this video, we are going to define Hausdorff topological space and discuss a couple of examples.
If you like the video, please help my channel grow by subscribing to my channel and sharing my videos.
Thank you for watching!
Maksym Zubkov
[email protected]
-~-~~-~~~-~~-~-
Please watch: "Real Projective Space, n=1"
https://www.youtube.com/watch?v=2ottRuDA5WA
-~-~~-~~~-~~-~-
Twitter: https://www.twitter.com/maksym_zubkov
Instagram: https://www.instagram.com/m.zubkov_
Math Blog: https://maksymzubkov.com
In this video I summarize Furstenberg's topological proof of the infinitude of the prime numbers. #math #stem #proof #numbertheory #primenumber #topology
This is a second video on the study of Topological Spaces.
Here, we try to learn how to determine whether a collection of subsets is a topology on X or not.
You can find the link to the previous video here;
https://www.youtube.com/watch?v=snMGzSffMDM
N.B. Some authors use the name Top for the category with topological manifolds as objects and continuous maps as morphisms.
As a concrete category
Like many categories, the category Top is a concrete category (also known as a construct), meaning its objects are sets with additional structure (i.e. topologies) and its morphisms are functions preserving this structure. There is a natural forgetful functor
to the category of sets which assigns to each topological space the underlying set and to each continuous map the underlying function.