Why is a discrete topology called a discrete topology?

Question

I'm not looking for the definition of discrete topology given in textbooks; I'm wondering why the word 'discrete' was chosen.

I mean, the concept of discrete topology is built up from sets, which are built from objects--which are discrete. So, if we're looking for a word to differentiate power sets as topologies from other topologies, and we use the adjective 'discrete' to accomplish that differentiation because the power set is composed of discrete objects--then, by similar reasoning, couldn't we call all topologies on sets 'discrete'.

Because they're built from discrete objects and compositions, too. All of them. All topologies.

There must be some other reason we call discrete topologies 'discrete'. What is it?

Answer 1

The discrete topology has a topological structure which perfectly reveals the discrete nature of the underlying set of points

You can consider a set to be a discrete collection of objects. To a given set $X$, you can assign a variety of topologies. Let's argue for the appropriateness of calling this particular topology "discrete".

1. The discrete topology is the finest topology—it cannot be subdivided further. If you think of the elements of the set as indivisible "discrete" atoms, each one appears as a singleton set. You can effectively "see" the individual points in the topology itself.

   Contrast this with the indiscrete topology, consisting only of $X$ itself and $\varnothing$. This topology obscures everything about how many points were in the original set. It fully agglomerates the points of the set together.

2. Revisiting this point, it's sometimes helpful to think of topologies as obscuring or blurring together the underlying points of the set. Topologies are all about nearness relations: points in an open set are in the vicinity of one another. If there are two points that never appear alone in an open set, those points are topologically indistinguishable. From the perspective of the topology, they are so close as to be identical.

   It is therefore remarkable that the discrete topology has no indistinguishable points. The discrete topology is the topology that obscures nothing about the underlying set. Each point in the set is clearly highlighted and distinguishable and recoverable as an open singleton set in the topology.

3. If you think of topologies that can arise from metrics, the discrete topology arises from metrics such as $d(x,y) = \begin{cases}0 & x=y\\1&x\neq y\end{cases}$. This metric "shatters" the points $X$, isolating each one within its own unit ball. In such a space, the only convergent sequences are the ones that are eventually constant; you can't find points arbitrarily close to any other points. Because points are isolated in this way, it makes sense to call the space "discrete".

4. Every function from a discrete space is automatically continuous. I'd argue that for this reason, the discrete topology is the one that best "represents" $X$ in topological space. Indeed, in many ways the nature of a set is characterized by its functions, and the nature of a topological space is characterized by its continuous functions.

   So, note that if $T$ is any topological space, there's a natural bijective correspondence between functions $f:X\rightarrow \mathsf{set}(T)$ and continuous morphisms $g:\mathsf{discrete}(X)\rightarrow T$. For every function on $X$, you can find a continuous function on $\mathsf{discrete}(X)$, and given any continuous function on $\mathsf{discrete}(X)$, you can uniquely recover a function on $X$.

   The discrete topology best represents the structure of the set $X$ which, as you say, is discretized into individual points.

5. Throughout abstract algebra, isomorphisms describe which structures are "the same". A topological isomorphism (a homeomorphism) between two topologies says that they are essentially the same topology. An isomorphism of sets is just a bijection; it says that the sets contain the same number of elements.

   Continuing the discussion of functions above, two discrete topologies are topologically isomorphic (homeomorphic) if and only if their underlying sets are isomorphic as sets (bijective). Put casually, this means that the discrete-topology-creating process maintains the similarity and differences between the underlying sets: discrete topologies are the same if and only if their underlying sets are.

   This is all the more important when we realize that sets are the same when they have the same number of points. Hence discrete topologies are the same when (and only when) their underlying sets have "discrete points" in the same quantity. You can count the points in a discrete topology through isomorphisms, and the discrete topology is the only topology for which this is possible.

Answer 2

Discrete in this sense means separated by a significant distance. In the usual topology of the reals every open set contains a lot (uncountably many) of points. In the discrete topology every point is an open set, so it is like the integers on the number line-each point is far away from every other point. Once you do that every subset of the space is an open set, so the topology is determined up to isomorphism by the fact that it is discrete and the number of points in the set. That is why we speak of "the discrete topology" on a set.

Answer 3

There is a resemblance between the set of integers which we call it discrete and the discrete topology.

Every point is closed and every point is open in discrete topology. It is called discrete because every single point is a component by itself. every point is an open neighborhood of itself.