In this article I want to talk about introduction to topology and I’m going to introduce the basic concept and I will give you the overall picture of the apology and why do we study topology? Okay, so let’s start with why doing in topology and let’s for example take us here And also, we’ll take a doughnut or another electors and Then I’m going to ask a simple questions. Are they the same or not? And the answer like is not why not because he can seize its fear there is not hope so just you can sink one two dimensional hole which because this is like as two and Sphere is just a boundary of the sphere.
So Sphere is bound emptiness inside And this absence effect is two dimensional but for the torus it you can see it always bounce like emptiness inside which is bounded by Let’s say like this section, but also we have a hole over here So the pelagic Allah speaking these two surfaces are different, but I was able to distinguish the surfaces because I was just looking at them So what apology do the apology developed tools that allows to? distinguish these two surfaces for example One of such tools are called algebraic topology when you use algebraic structure to distinguish.
The two surfaces are not the same and one of the such Absurd object is called fundamental group and in my future video. I’m going to show that fundamental group of as to Its trivial or in other words equals to zero But fundamental group of a the torus or in other words be indicated by t2 It’s not trivial and equal to direct Equal to Cartesian product of Z And we can see since Fundamental groups are different.
That means that these two surfaces are different Now, let me just tell you what the fundamental group is. So fundamental group is basically saying that Let’s take some any point in my space in other forms of my surface and I will draw a loop the Fundamental group of seta tube is trivial because you can see for any loop that you take You always can trim this loop to a point so you take a loop and you shrink and you get a pot and this is true for any appointments fear and for Angelou, but Here on a torus, you can see that if I will take for example exist Lou Then I can shrink this loop To tow it down to a point, but if I’m going take this loop It doesn’t matter how I’m going to move this loop around the torus I will never be able to shrink this point to a Down and You also can see that I have two of such kind I have this loop or I have loops that goes around this empty space around the circle and That’s why I have Z 2 because 1 z generated by this law and as RG,
You know in two phases So this type of surface are really difficult and let me show what path leads ask to be able to use these tools and what do we need to start in topology if you want to be able To apply these tools and to be able to distinguish like two surfaces like tube manifolds topological spaces that there is a same or not So in order to show this how to distinguish like big ones let first draw small ones Let’s first let’s say draw a circle, which is like just this one and a line interval And I want what about to say about I want to say that this? To like Spaces like this is two objects right does the same?
So I want to show that there are not Harming am orphic And what does it mean homeomorphic it means like this exists and mapping within two spaces and This mapping you can think about that you take one space and you want to deform it You’re not allowed to break it You’re not allowed to cut it and reattach the only like transformations you can do You can squeeze it you can make it smaller bigger and you can see if a vortex the circle will try to rotate I will Never make the circle to be aligned But you will ask the wrong way because I can have I can take my circle and just make it smaller and smaller so I kind of Map my circle into editable. Yes, but being homeomorphic means every time you do this transformation your transformations by ejected so you never overlap your points and If you’re the shtetl dispatch active means then like this map must be by ejected So we want to study some maps and this maps Required some space.
So the apologies the first things that we study in topology, which study topological space So topological space we just introduced like an abstract mathematical Space and for this space we’re going to have and we want differ But on this space we want to define such mappings, but if you want to define mappings with certain properties We want to discuss after the pelagic of space for this continuous map After we’re going to discuss continuous Maps the next thing we want to discuss homeomorphism And when we have powerful Murphy is B, so it’s saying that this new space is not Hermia orphic so in terms of topology,
It means if I will have Map between s 2 2 interval Then I’m saying that F is homomorphism If I have two properties first f is bijective Second one f is continuous and the third one inverse off is continuous so By introducing this tree concept on the our let’s a third video you will be able to understand What is what does it mean to be homeomorphic? And let’s show me the last thing let me show that these spaces are not hammer homeomorphic Let’s say the four things that you want to start if we want to start a connectedness And fuckin Agnes means that for example circle and Unit interval is one connected is they have only one connected component? Why because you can see you cannot get of separate your space into two disconnected component and again going to give you definitions later It’s a cool property about homie amour homomorphism that some topological like structure some topological feature of our space are going to be saved under homomorphism so if.
This like to space is homeomorphic it means like If they like connected that you can see that when you met one space to another one the another space must be connected If one space is bounded like a compact another space also must be compact so compact it means like closed and bounded But here you can see if I’m going to remove here one point since f is bijective, I’m going to remove only one point on my unit interval and let’s say that I’m not removing like boundary point and I can do this because if I remove boundary points and you just remove another point and then you can see that if there were Homeomorphism it with s2 in I or in other words if they’re more homeomorphic Then by removing one point on the from a morphism They must have the same number of connected components and you can see indeed.
This circle can be transformed just Into this unit interval and it has a one connected component But this interval has two connected components so it means these two spaces are not from a orphic and These are basic tools that allow us to distinguish two spaces, but sometimes our spaces are really difficult So in this case we want to study More advanced topology, which is called algebraic topology where we’re going to use algebraic structures Which are fundamental rules homolog caromed ology groups. Yeah. So this is like a small introduction to the apology.