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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?

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    $\begingroup$ By your reasoning, everything in mathematics is "discrete". $\endgroup$ – Eric Wofsey Jan 21 '18 at 4:33
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    $\begingroup$ Each element is its own little island. That makes it discrete. In the real numbers, for example, you're right that each real is an individual element. But they're all jammed together. You can't draw a little circle around a real number without including lots of other nearby ones. Whereas in the integers, say, you can do that. So the integers are a discrete set. $\endgroup$ – user4894 Jan 21 '18 at 4:35
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    $\begingroup$ From wikipedia: "In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. The discrete topology is the finest topology that can be given on a set, i.e., it defines all subsets as open sets. In particular, each singleton is an open set in the discrete topology." $\endgroup$ – user 170039 Jan 21 '18 at 4:35
  • $\begingroup$ See also this answer. $\endgroup$ – user 170039 Jan 21 '18 at 4:39
  • $\begingroup$ @user170039 - So, is it possible then to have a discrete topology on the set of all real numbers? I mean--sure, the topology would have uncountably many subsets of the reals, but conceptually a discrete topology on the reals is possible, no? $\endgroup$ – StudentsTea Jan 21 '18 at 4:54
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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.

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    $\begingroup$ Dude. This is great. I've got goosebumps. Thank you! ( I see you're editing it now. I'll come back in a few minutes to read it again. ) Thank you! $\endgroup$ – StudentsTea Jan 21 '18 at 5:02
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    $\begingroup$ Glad to help! Let me know if I can clarify or add anything else. :) $\endgroup$ – user326210 Jan 21 '18 at 6:49
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    $\begingroup$ Correction: It is not true that all indiscrete topologies are isomorphic; indeed they are homeomorphic iff the underlying set cardinalities are the same, just as with the discrete topology. This is becase a homeomorphism is a bijection on the underlying set, by definition. You need a looser notion, like homotopy equivalence, to equate $2$ and $3$ as indiscrete topological spaces. $\endgroup$ – Mario Carneiro Jan 21 '18 at 7:54
  • $\begingroup$ So, I keep seeing in all these explanations of discrete topologies the concept of metrics--this idea that sets and objects in topologies are comparable and, only when our metric ( or topology? ) fuzzes out details of the underlying domain do we stop calling the topology discrete. This thought is leading me to wonder--are all topologies married to metric spaces? If you have a topology, do you necessarily also have a metric space? $\endgroup$ – StudentsTea Jan 22 '18 at 6:57
  • $\begingroup$ That's a great question! Every metric space gives rise to a topology, but there are some topologies whose nearness relations can't be associated with any consistent metric. An example is the "line with two origins". $\endgroup$ – user326210 Jan 22 '18 at 7:14
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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.

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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.

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