Why does topology run on the two rules no cutting and no gluing?

Question

Topology uses the rules no cutting and no gluing. (It is a way of saying 'conservation of holes'.) Well out of all the rules it could've used, why does it conserve holes? I mean it could've been conserve volume, conserve mass or conserve surface area but the creator of topology chose conservation of holes...why?...plus topology doesn't apply in real life situations at all (as far as I know). So it would be great if you could explain to me why topology uses conservation of holes ...and also tell me some real life applications of topology...

"plus topology doesn't apply in real life situations at all" lol — , Oct 11 '20 at 02:25
Similar questions have been discussed on this site a number of times (see here, here, here, and here). — Stahl, Oct 11 '20 at 02:29
Additionally, the definition of a topology doesn't talk about "cutting" or "gluing," and those concepts might be more difficult to formalize than you imagine. The most common definition of a topological space is in terms of open sets. And with that in mind, the definition of a topology means that an arbitrary topological space is not "rigid" enough to define things like volume, surface area, or mass. The extra data that would allow you to define things like this is some sort of "geometry," and I would loosely say that such a space would be a geometric space, not just a topological space. — Stahl, Oct 11 '20 at 02:32
There are many applications of topology to various areas of mathematics, but some more "practical" applications are in physics, machine learning, or even politics: see here, here, here, and here to start. — Stahl, Oct 11 '20 at 02:34

score 2 · Answer 1 · answered Oct 11 '20 at 02:52

First of all, the "no cutting and no gluing" slogan is not how topology is defined: it's an attempt to give an intuitive description of the notion of homeomorphism, or "topological sameness." But that notion is defined in a more technical way (namely the existence of a bijection which is both open and continuous). So you shouldn't ask why topology focuses on cutting and gluing, since that's not really how it starts.

Topology can be thought of as starting with the question: "Which notions from analysis/geometry/etc. can be defined purely in terms of openness/closedness of sets and continuity of maps?" For example, the extreme value theorem says that every continuous function $C\rightarrow\mathbb{R}$ has a maximum and minimum whenever $C\subseteq\mathbb{R}$ is closed and bounded. This a priori uses the notion of boundedness, which is not just about openness/closedness/continuity; is that essential? It turns out that it's not: the notion of compactness is defined purely in terms of (families of) open sets, and is exactly the property which makes the EVT work.

From that starting point we whip up the abstract notions of topological space and continuous map. This provides us with a machinery not just applicable to things like $\mathbb{R}^n$, but also weirder objects like the Zariski topology which is important in abstract algebra but is very unlike the nice spaces we're used to from ordinary geometry (e.g. it's not Hausdorff, so a fortiori doesn't come from a metric).

Now "topological sameness" (= homeomorphic-ness) is a very coarse relation: there's a lot you can do to a geometric object without changing its topology. However, conversely this means that topological features and changes in topology are significant. Similarly, things you can prove which just depend on the topology of an object are highly robust. Paying attention to the topology of an object, perhaps how that topology changes over time, gives us a certain "big picture" focus. This can be quite useful - see e.g. applied algebraic topology (and persistent homology in particular).

It's also worth noting that topology is by no means the only game in town. We can look at lots of different kinds of "geometric sameness," e.g. metric geometry, Riemannian geometry, symplectic geometry, ...

score 1 · Answer 2 · answered Oct 11 '20 at 02:25

Other branches of mathematics (including geometry) study transformations that preserve straight lines, or areas, or volumes. (Look up "Affine geometry.")

Conservation of holes is not the postulate or condition, but instead the consequence of the conditions of no cutting and gluing.

There are many practical applications of topology, including many from knot theory (a sub-branch) that have even been applied to DNA. Topology governs a large number of differential equations that apply to the real world. In physics, the topological structure of space-time (including its dimensions) is of profound consequence. Here's a starting point.

score 0 · Accepted Answer · answered Oct 11 '20 at 05:32

Actually 'no cutting and no gluing' is not exactly the intuition you should have about topology. For example, a trefoil knot is clearly homeomorphic to a circle, but because of the difference of embedding, they can't 'transform' to each other in $\mathbb{R}^3$, the 3-dimensional space we are in.(This involves a fascinating branch called knot theory) For other examples, cut a band, turn an end for $360^\circ$ and glue them back together, you get a shape homeomorphic to ordinary rings, but you cannot 'transform' it into a ring, intuitively.

On the other hand, topology has its language to describe cutting and gluing decently. For example, glue a pair of antipodal points on a sphere together, and its 2-dimensional homotopy group stays the same. (though this is not intuitive)

The point is that topology space is a $\textbf{intrinsic}$ concept, which means, it is not about a shape inside a space, but about the shape as a space itself, therefore the concept of transformation should be considered in a more formal and less intuitive way. The history has shown many times that human intuition is limited, and when exploring mathematics we should always stay humble and careful.

Why does topology run on the two rules no cutting and no gluing?

3 Answers3