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I am very new to topology. I want to know why we need topology in mathematics. What is its significance? Without topology which mathematics concept cannot be explained. What are its real-life applications? I know the basic definitions of topology.

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marked as duplicate by Aloizio Macedo, Martin Sleziak, N. F. Taussig, Kamil Jarosz, SchrodingersCat Jan 18 '16 at 14:57

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Here's a possible answer to "why study point-set topology": General topology tries to abstract out the notion of "distance" on $\Bbb R$. Indeed, a large part of point-set topology studies metric spaces - it turns out to define continuity, we don't need to know much about real numbers. Given any set $X$, if there is a map $d : X \times X \to \Bbb R$ satisfying $d(x, y) = 0$ iff $x = y$, $d(x,y) = d(y, x)$, the triangle inequality and $d$ is non-negative, then this extra structure on $X$ is sufficient to study some amount of "analysis" with it. This enables us, for example, to do analysis with graphs, which are thoroughly discrete objects.

Abstract topological spaces takes this to a greater height - you actually don't need a metric either. Just defining which elements of $X$ are "close" is enough, you don't need to know how close they are. That said, you pick a subset $\tau$ of $\mathcal{P}(X)$ satisfying some desirable properties ($\tau$ is closed under arbitrary union and finite intersection, e.g.), where elements of $\tau$ are the subsets of $X$ consisting of elements "close" to each other. The members of $\tau$ are said to be the open subsets of $X$. This is what a topological space is.


As for real life applications, there are quite a few, but I am not aware of any "elementary" examples. In data analysis, persistent homology is quite an active branch of study. Roughly, if you have a data set representing, say, a fingerprint, and you want to recognize it, then you'd associate some numbers ("presistent betti numbers") to it as follows: If $X$ is a collection of data points on $\Bbb R^2$, let $X_\delta$ denote a $\delta$-thickening of $X$, i.e., union of neighborhoods of radius $\delta$ of the points in $X \subset \Bbb R^2$.

If you vary $\delta$, the topology of $X_\delta$ will change, but if you vary $\delta$ at a certain speed, you can look how much time it takes for $X_\delta$ to change it's topology. For example, you could look at the betti numbers of $X_\delta$ and see how long they stay the same. Among the betti numbers $b_n$ of $X_\delta$ the one which stays same the longest amount of time is called the persistent betti number. For example, if our data set is a bunch of point cluttered around the unit circle, $b_1 = 1$ would stay for a long time.

This "persistance" data lets you identify different data sets.


For an example of topology being useful in other branches of math: One of the most frequently used facts in complex analysis is the Jordan curve theorem. It says that every simple closed curve $\gamma$ in $\Bbb R^2$ divides $\Bbb R^2$ in two regions, that is, $\Bbb R^2 \setminus \gamma$ has two connected components. Furthermore, each of the two connected components are topological disks, i.e., homeomorphic to $D^2$.

The reason why this is remarkable is that no smoothness assumption on $\gamma$ is made. It can be thoroughly non-rectifiable, like a Koch snowflake, and nowhere differentiable. Is it obvious that the Koch curve divides the plane into two regions? Apparently not. Another thing to ponder on: it's false that every topological sphere in $\Bbb R^3$ bounds a topological disk on both sides (although it is true that it divides $\Bbb R^3$ into two components). This is the Alexander horned sphere. The point is to emphasize that this is actually a big theorem.

The proof of this, however, is done using homology (i.e., algebraic topology).

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The already present answers covered much of the topic, so I am going to give a very simple example from classical mechanics as to why is this concept useful.

Imagine a classical mechanical system composed of two particles, and we do not have any constraints. The configuration space of one particle can be identified with $\mathbb{R}^3$. The configuration space of the entire system is then $\mathbb{R}^3\times\mathbb{R}^3\simeq\mathbb{R}^6$.

Obviously, since $\mathbb{R}^3$ represents the physical euclidean space, it has a metric structure. If the standard coordinates are cartesian, then the distance between two points, $x=(x_1,x_2,x_3)$ and $y=(y_1,y_2,y_3)$ is $\|x-y\|=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2}$.

On the other hand, if you consider the entire configuration space, you COULD define a metric structure (for example in this case, take the orthogonal direct sum of the two $\mathbb{R}^3$s and you have a metric on $\mathbb{R}^6$, but it physically makes no sense. If you give two separate configurations of this two-particle system, it does not make physical sense to ask "how far apart these configurations are".

On the other hand, it absolutely makes sense to ask if a sequence of configurations converge to a certain configuration or not, in fact, this concept of convergence is absolutely necessary if you want to do any calculus on this system.

Topology allows you to define convergence and continuity without having any actual distance function. As my example illustrates, in many cases it is needed to have convergence and continuity, but any distance function that might induce them is superfluous and contains unneeded information.

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If you want an application: Using Kuratowski's theorem you can check if a planar graph can we drawn in such a way that its edges don't intersect. I guess this can be used when designing electric circuits and computer parts.

https://en.wikipedia.org/wiki/Kuratowski%27s_theorem

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Topology has many applications since it is concerned with the properties of space that are preserved under continuous deformations. I think some really important properties in topology are connectedness and compactness.

Also, you can take a look for real life applications in wikipedia: https://en.wikipedia.org/wiki/Topology#Applications

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