What Is Topology: Definition & Examples

[En_lizard]

what is it?

 

[matt grime]

A very large area of mathematics that would struggle to be summed up in one line.

In its most elemental description we should define a topology on a space X.

A topology is a collection of subsets of X satsifying certain properties. These are labelled as open. The complements of open sets are closed.
The rules are simply that if we take a finite number of sets labelled open and intersect them we must get another set labelled open. If we take any number of open sets and take their union the resulting set must be also labelled open.

Remarks: it is perfectly acceptable for a subset of X to be neither open nor closed. It is also possible for a subset to be both open and closed.

Example: on the real numbers define O to be open if for all x in O there is an open interval (a,b) containing x lying wholly in O. Try showing this is a topology. (It is the most common type you'll meet)These simple definitions allow us to shed light on a huge amount of mathematics.

In particular this simple allocation of some subsets as open, some as closed (and some are neither) allows us to completely overhaul calculus: no more epsilons and deltas. A function f is continuous if and only if for each open U $$f^{-1}(U)$$, which is by definition the elements mapping to U, is open.

Example: take the natural numbers, {1,2,3,4...} and the integers {0,1,-1,2,-2,3,-3,...} and give these sets the discrete topology (a set may have many topologies). The discrete topology always exists for any set and declares that every set is open (and closed too).

The inclusion of the naturals into the integers is now a continuous map.

It turns out we can classify surfaces (things like a sphere, or a torus aka bicycle inner tube) simply by the 'number of holes' in them, purely by looking at how the open sets glue together. A sphere has no holes, the inner tube one.

We can also reduce difficult questions about these spaces and sets to ones in algebra (indeed the classification of surfaces does just that) via algebraic topology.

This is perhaps what some might label 'rubber sheet' geometry. The idea is that we should view topological spaces as the same if there is a continuous bijection f with continuous inverse between them, f is called a homeomorphism, and two spaces linked by a homeomorphism are homeomorphic. Topologically we cannot distinguish between these things, even though they may look very different to the naked eye. For instance topologically, the integers and the natural numbers are equivalent with the discrete topology we gave earlier. To see this note that if we give a set X the discrete topology then any function from f must be continuous because we declare every set to have the label open. This means that all we need to do for the integers and the naturals is write down a bijection because then it is automatically a homeomorphism. You should already know that these two sets are in bijection, indeed the way i wrote them out shows a bijection failry clearly.

Actually we find that the classification up to homeomorphism is often far too strong, ie topologically we can't distinguish between things that aren't even homeomorphic, and we have to relax to something slightly more obtuse called homotopy. Homotopy roughly means that we can squish things down but don't have to worry about going back. For instance consider the shapes of letters o and d, these are not homeomorphic shapes - we can prove this; consider removing the point where the upright of the d meets the loop, this disconnects the shape into two separate shapes, but removing any point of o leaves only one shape behind - but they are homotopic because we can squash the upright down to a point. What this means is many of the algebriac gadgets we invent to test whether two shapes are similar can't differentiate between these two objects.

Topology is inherent in what we now call (algebraic) geometry, one of the results of which is that there is no way to comb the hair on a tennis ball smooth. (Any vector field is not smooth.)

I once heard it described as the analysts attempts to become algebraists.

 

[HallsofIvy]

In a nutshell, topology is essentially the mathematics behind the concept of continuity in its most general, abstract form.

 

[JasonRox]

That's why I voted for matt grime.

I'd keep a library of my postings, which the site does, so you can just copy and paste for identical/similiar questions, especially those a/0 questions.

 

[fourier jr]

yes, recall the definition of continuous function for metric spaces:
If $$(M, \rho)$$ and $$(N, \sigma)$$ are metric spaces, a function f:M -> N is continuous iff for each $$\epsilon > 0$$ there is some $$\delta > 0$$ such that $$\sigma(f(x), f(y))<\epsilon$$ whenever $$\rho(x,y) < \delta$$

& remember that an $$\epsilon$$-disk about a point x is the set U(x) = {$$y \in M | \rho(x,y)<\epsilon$$}, so a set is open if every point in it has an $$\epsilon$$-disk. using this idea we can modify the original definition of continuity to the following:
$$f:(M, \rho) -> (N, \sigma)$$ is continuous @ x in M iff for every $$\epsilon > 0$$ there is some $$\delta > 0$$ such that $$f(U(x)) \subset U(f(x))$$

the open sets in a metric space M have the following properties (try it with disks, epsilons, etc):
i) any union of open sets is open
ii) finite intersections of open sets are open
iii) the empty set and the whole space M are both open

but it turns out that you don't really need the concept of distance. if you have a set & a collection of subsets of the set that satisfy the 3 properties above, you get a topological space, which is what topology is all about. if you ignore metrics & distances, a further modification of the definition of continuity (the 'topological' definiton) goes like this:
a function f:M -> N is continuous @ x_0 in M iff for each open set V in N containing f(x_0) there is an open set U in M containing x_0 such that $$f(U) \subset V$$ & there's no mention of any kind of distance, just open sets.

 

[En_lizard]

so that means topology! thanks alot!