What is the usefulness of topology in understanding functions?

[kent davidge]

I was asking to myself what is the usefulness of a topology. I'd thought this question before and couldn't find results on the literature, perhaps I was not searching with the right terms.

So I started thinking that maybe a topology is a way of defining the domain, codomain and image of a function?

For this I imagined the following situation. Let the set be $\mathbb{R}$ and define a metric $d$ on it such that $d(x,y) = 1$ if $x \neq y$ and $d(x,y) = 0$ if $x = y$. Such metric induces the topology $\tau = \{ \emptyset, \mathbb{R} \}$, correct?

Does that mean only functions whose domain is either $\mathbb{R}$ or $\emptyset$ can be defined on such topological space?

Ok, probably I wouldn't have needed to go too far... but that scenario was what I thought in my attempt to understand this.

 

[fresh_42]

I was asking to myself what is the usefulness of a topology. I'd thought this question before and couldn't find results on the literature, perhaps I was not searching with the right terms.

So I started thinking that maybe a topology is a way of defining the domain, codomain and image of a function?

No. The corresponding functions to a topology are the continuous functions. So the topology determines which functions are continuous, and which are not.

For this I imagined the following situation. Let the set be $\mathbb{R}$ and define a metric $d$ on it such that $d(x,y) = 1$ if $x \neq y$ and $d(x,y) = 0$ if $x = y$. Such metric induces the topology $\tau = \{ \emptyset, \mathbb{R} \}$, correct?

No. This metric is called the discrete metric and all sets are open (and closed), because $\{\,x\,\} = \{\,y \in \mathbb{R}\,|\,d(x,y)< \varepsilon\,\}$ is an open neighborhood of $x$. Therefore all functions defined on subsets of $(\mathbb{R},d)$ are continuous.

Does that mean only functions whose domain is either $\mathbb{R}$ or $\emptyset$ can be defined on such topological space?

No. If at all, the question is whether a function is continuous or not, and not whether it is defined. But in this special case, all functions are continuous.

Ok, probably I wouldn't have needed to go too far... but that scenario was what I thought in my attempt to understand this.

The discrete metric induces the finest topology, not the roughest (?! - not sure what the correct opposite is in English).

But you can simply define such a topology by $\{\,\emptyset\; , \;\mathbb{R}\,\}$ without any metric and consider the question, which functions are continuous.

 

[member 587159]

the roughest

The term is correct, although I still prefer to call those topologies the "largest"and "smallest" (for the inclusion).

 

[WWGD]

I think it is fair to say that a topology is a/the machinery need to be able to talk reasonably about continuity and other notions such as connectedness that are considered to be topological properties of a space.

 

[FactChecker]

I think it is fair to say that a topology is a/the machinery need to be able to talk reasonably about continuity and other notions such as connectedness that are considered to be topological properties of a space.

I agree. If one wants to say that a doughnut and a coffee cup are similar, but different from a sphere, then topology is the way to do it.
For an interesting (IMHO) and easily understood discussion of topology and the shape of the Universe, watch .
Another application is the theory of knots.
(see )

 

[mathwonk]

"coarsest" is another candidate.

 

[fresh_42]

The term is correct, although I still prefer to call those topologies the "largest"and "smallest" (for the inclusion).

Seems, it is not: https://en.wikipedia.org/wiki/Comparison_of_topologies

 

[FactChecker]

@kent davidge , It is difficult to know how or at what level to answer a question like this. There is no indication of your background in mathematics because your profile is hidden.

 

[member 587159]

Seems, it is not: https://en.wikipedia.org/wiki/Comparison_of_topologies

You are right. I googled on 'roughest topology' and some results popped up so I assumed it was correct! Apologies.

 

[Stephen Tashi]

I was asking to myself what is the usefulness of a topology. I'd thought this question before and couldn't find results on the literature, perhaps I was not searching with the right terms.

We should distinguish between "point set topology" versus "algebraic topology".

The general idea that "point set toplogy" is useful as an abstract way to state many important properties of metric spaces (and other types of spaces) is correct. If you use the concepts of point set toplogy, you can understand many results from real analysis, n-dimensional calculus, calculus on manifolds etc. without considering the special properties of the metrics defined on such spaces.

Your conception of point set topology itself needs fixing. A "topology" involves a set of "open sets". This set of open sets is usually an infinite set, so it's best to think of "open set" as being defined as a general type of set. (e.g. a union of open intervals in the case of real analysis). The general concept of a continuous function $f$ is that it makes the inverse image of any open set also an open set. It takes some effort to see why this general concept is equivalent to an epsilon-delta definition of continuity used in the calculus of real valued functions of 1 real variable and definitions of continuity used in other specific contexts. However, if you study a lot advanced mathematics, it's worth doing this. In fact, I'd say it's essential that you do this.

 

[kent davidge]

@kent davidge , It is difficult to know how or at what level to answer a question like this. There is no indication of your background in mathematics because your profile is hidden.

Oh, I didn't know that my profile is hidden. My background is Calculus 1 and 2, and Introductory Linear Algebra, as I'm on my second year of a Bachelor Degree Physics course.

The general concept of a continuous function fff is that it makes the inverse image of any open set also an open set. It takes some effort to see why this general concept is equivalent to an epsilon-delta definition of continuity used in the calculus of real valued functions of 1 real variable and definitions of continuity used in other specific contexts. However, if you study a lot advanced mathematics, it's worth doing this. In fact, I'd say it's essential that you do this.

It's interesting that before reading the latest replies here, I came across these concepts when looking this topic on the web. It makes a lot more of sense for me now.