EDIT (to clear up controversy and misunderstandings caused by my poor wording): Historically, Riesz's efforts to try and make rigorous a notion of a "continuous space" (as opposed to "discrete ones") were part of some of the movements in mathematics which lead to the establishment of the axioms of topology.
Some people tout topologies as being "the axiomization of the notion of a continuous space". However, they are clearly too general, as I argue below.
What is an appropriate level of axiomization to make rigorous a notion of "continuous space"?
Why do we allow either discrete or trivial topologies in the definition of topology? (if we want it to define "continuous spaces", otherwise there is no issue)
More generally, why do we allow points to be open (the existence of isolated points) or the failure of the T0 axiom (existence of topologically indistinguishable points)? (if we want it to define "continuous spaces", otherwise there is no issue)
Both scenarios could be prevented by addition of the following axioms to the definition of topology:
- No point is open
- Every point has a unique neighborhood system
This question is a follow-up to my previous question: Why study non-T1 topological spaces?
The historical motivation for topology, as far as I have read, was to give a rigorous notion to the intuition of a "continuous space". The three axioms commonly used are equivalent to those proposed by Kuratowski, Riesz, and Hausdorff, who were trying to axiomatize "continuous space".
But discrete spaces are the exact opposite of that, and in general allowing isolated points allows the possibility of spaces with "discrete components". It also makes the definition of "limit" unnecessarily complicated. The discrete topology is also equivalent to the power set, so is in effect not a new notion. Also every function from a discrete space is continuous, and any definition of a function at an isolated point is continuous, which goes counter to the expectation that the morphisms of a given category should be "special" functions. Basically discrete topologies gains us no new insights compared to elementary set theory.
Moreover, points not having unique neighborhood systems lead to all sorts of pathologies (for example here: Non-T1 Space: Is the set of limit points closed?), and even the Zariski topology, which is the most pathological commonly used topology of which I am aware, is still in general $T0$.
Finally, any non-$T0$ space should be homeomorphic (I think) to the quotient of the space under the equivalence relation " $x \sim y$ if and only if $x$ and $y$ are topologically indistinguishable".
This question is similar in spirit to this one, but to me it is very clear why we would want to consider non-metric spaces, at least as someone who is interested non-metricizable spaces like those which can occur in functional analysis. I am aware of the notion of continuum, but since the definition requires the space to be metric, and focuses more on compactness than separation axioms and connectedness rather than path-connectedness, I believe it to be the incorrect axiomization of a "continuous space". The existence of indecomposable continua lends credence to this notion, in my view.
However, it seems clear to me that the proper axiomization of a "continuous space" must lie somewhere in between the notion of "metric space" and "topological space", as the former is far too restrictive, and the latter far too general. I would enjoy your thoughts on the matter.
(I want to better axiomize the notion of "continuous space" so as to better facilitate the study of stochastic processes, not that this is directly relevant to the immediate question at hand.)
At least Riesz appeared to have been clearly interested in defining some notion of "continuous space" or "continuum" (excerpt from "History of Topology" on google books (https://books.google.com/books?id=7iRijkz0rrUC&pg=PA212&lpg=PA212&dq=riesz+topology+axioms&source=bl&ots=B_xcnG6StL&sig=EajpOCr3XRUtA9hwqpLwArKLmoY&hl=ru&sa=X&ved=0ahUKEwj3pMPs9ZvNAhVBGFIKHcXmB28Q6AEIJzAC#v=onepage&q&f=false):
In a footnote Riesz criticisms the way in which philosophers have dealt with notions like continuous and discrete and he repeats Russell’s remark about the followers of Hegel: «the Hegelian dictum (that everything discrete is also continuous and vice versa) has been tamely repeated by all of his followers. But as to what they meant by continuity and discreteness, they preserved a discrete and continuous silence; […]» (Riesz ). The relation of our subjective experience of space and time and mathematical continua is described by Riesz as follows. Mathematical continua possess certain properties of continuity, coherence and condensation. On the other hand, our subjective experience of time is discrete and consists of countable sequences of moments. Systems of subsets of a mathematical continuum can be interpreted as a physical continuum when subsets with common elements are interpreted as undistinguishable and subsets without common elements as distinguishable. Rise [147, p. 111] is an interesting paper in which Riesz, who had read Frechet’s work and appreciated it, developed a different theory of abstract spaces, based on the notion of «Verdichtungsstelle», i.e. «condensation point», or as we will translate «limit point». In his theory Riesz succeeded in deriving the Bolzano-Weierstrass Theorem and the Heine-Borel Theorem. We will not discuss this paper. We will restrict ourselves to a shorter paper that was presented by Riesz in 1908 at the International Congress of Mathematicians in Rome. In that paper, «Stetigkeit und Abstract Mengenlehre» (Riesz ), concentrates on the characterization of mathematical continua. We will briefly describe some of the ideas the Riesz describes in the paper. As we said, Riesz’s basic notion is the notion of a limit point (Verdichtungsstelle).
I'm not asking anyone to agree with Riesz's basic goal (to get a rigorous distinction between "continuous" and "discrete" spaces or time).
To those who might object that the standard three axioms are all that are necessary to define the notions of connectedness and compactness, I have several responses:
Connectedness actually isn't that nice of a property. We could look at the topologist's sine curve or the infinite broom for examples, but for me the fact that there exists a countable, Hausdorff, and connected set implies that topological connectedness is not quite the intuitive connectedness from Euclidean spaces which we want to generalize for an arbitrary "continuous space". In my mind, path connectedness better satisfies this criterion, since every non-trivial T1 path-connected space is uncountable. Path connectedness does provide reasonable properties, although one can object that the definition is somewhat of a tautology ("a space behaves like the real line if it behaves like the real line") and requires the prior construction of the real line in order to define, rather than being constructible from first principles.
Compactness also isn't very nice unless we are in a Hausdorff space, since otherwise compact sets aren't even closed. Sure $T0$ isn't Hausdorff, but it's certainly a step in the right direction. (Hence why the distinction between "compactness" and "quasicompactness" is so prevalent in algebraic geometry where the non-Hausdorff Zariski topology so frequently comes into play).
For non-$T0$ spaces, the set of limit points isn't even closed, limits aren't unique or well-defined... the notion of limit so intrinsic to the idea of a "continuous space" clearly requires at least $T0$, if not even $T1$ or $T2$, to function even remotely similarly to intuition.
Related but different: Why the axioms for a topological space are those axioms?
The definition of metric space,topological space
What concept does an open set axiomatise?
Why do we require a topological space to be closed under finite intersection?
meaning of topology and topological space
Motivation behind topology
Why is discrete space ``discrete"
Discretizations of Differential, Geometric and Topological Notions
Origins of the modern definition of topology