# How to Axiomize the Notion of “Continuous Space”?

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.)

EDIT:

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 [147]). 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 [148]), 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:

1. 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.

2. 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).

3. 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.

• Comments are not for extended discussion; this conversation has been moved to chat. I deleted the whole lot, so check that chatroom, if you need to retrieve/ refer to any of them. In the end of the discussion I was detecting traces of attempts at (one-sided) insults. Please don't go there. – Jyrki Lahtonen Jun 13 '16 at 5:46

Perhaps topological manifolds are what you're looking for. Every connected topological manifold can be thought of as a "continuous space." If you don't demand connectedness, they can be "discrete." This eliminates examples like $\mathbb{Q}$, for better or worse.

Another possible solution is to just focus on "metrizable" topological spaces, or "completely metrizable" ones. The former allows $\mathbb{Q}$, the latter disallows it. Make sure to allow your metrics to take values in the interval $[0,\infty],$ otherwise your category of metrizable spaces won't have coproducts.

Also, Flagg spaces (he calls them "continuity spaces", but this is a poor choice of phrase) are a generalization of metric spaces in which every topology arises from a suitably chosen metric. So you might find that interesting; I think they're mainly used in computer science.

A general comment: if you're interested in the question of how much generality is needed in the definition of a topological space, its worth learning as much functional analysis as you can, and especially just spending a lot of time thinking about infinite-dimensional function spaces.

For a very different viewpoint on what the axioms of topology ought to be, take a look at the nlab page on convenient categories of topological spaces. The idea is that a good category of spaces should have nice categorial properties, like $\mathbf{Set}$ does. My personal feeling is that we need to generalize, rather than specialize: convergence spaces are imo a very natural and appealing framework in which to formalize topological intuition. You may also be interested in uniform convergence spaces.

• This is the best and most detailed answer so far in my opinion-- thank you so much for your help. I agree that convergence is an important property to generalize which we expect to get from continuous spaces; I think this is what Riesz, Hausdorff, and Kuratoski were trying to get at. Topological manifolds do look about what I am looking for, although they might be too restrictive, since I am unsure if local homeomorphisms to R^n should be just right or too much to ask of a continuous space (although the nice properties of Hausdorff and locally path connected spaces suggest the former). – Chill2Macht Jun 12 '16 at 14:22
• Anyway I will look more into all of your suggestions to see which best matches the notion of continuous space I have in mind; all sound like good candidates for a notion of continuous space imo – Chill2Macht Jun 12 '16 at 14:23
• @William, glad to be of use. I hope you find something reasonable. – goblin Jun 12 '16 at 14:29
• Apparently the category of Flagg's "continuity spaces" is equivalent to topological spaces, so I would agree with you that the name is a misnomer, just as "continuous function" is a misnomer for topological morphisms. math.stackexchange.com/a/1071730/327486 – Chill2Macht Jun 12 '16 at 18:05
• @William, just to be clear, the reason I said it was a misnomer is because morphisms of Flagg spaces are much more restrictive than continuous functions. For example, metric maps are much more restrictive than continuous functions. – goblin Jun 12 '16 at 18:19

There was a teacher of mine which used to restrict the definition of continuity to the points of the set which were limit points. Hence, asking if the function $f:\{0 \}\cup [1,2] \to \mathbb{R}$ was continuous was not meaningful to him.

I asked him "why are you restricting it?" He answered (paraphrasing, of course. I don't remember the exact words): "Because continuity is a concept related to the process of limits, it is not quite meaningful to attribute the word 'continuous' to a point to which nothing is converging to." This phrase is quite, quite similar to your "proper axiomatization of a continuous space", whatever that means.

Turns out that later in class he needed to use the continuity of a function which he was restricting. I asked promptly: "Why is this function continuous? You first need to prove that all points of your set are limit points." After some discussion, he conceded on his definition of continuity.

The bottom line is: A lot of times in mathematics, it so happens that grasping to one's psychological comfort only leads to unnecessary hindrance. Making more assumptions than needed will often lead to useless labor, and also to wasted time.

• Shouldn't one basically know a priori in most cases if all of the points of one's set are limit points, unless one is working with completely arbitrary abstract spaces? The statement is trivial for metric spaces for instance – Chill2Macht Jun 12 '16 at 3:44
• @William Limit points are points $p$ for which every neighbourhood intersects points different from $p$. By "this is trivial for metric spaces", do you mean that all points of a metric space are limit points? If so, not only is it not trivial, it is also not true. – Aloizio Macedo Jun 12 '16 at 3:50
• No, but for a connected manifold, or most reasonable spaces. Or the typical Banach/Hilbert/locally convex space. And most counterexamples can be embedded into spaces for which the claim is true. – Chill2Macht Jun 12 '16 at 3:55
• @William Do you realize how circular your argument is? It doesn't seem you want an answer, but rather it seems you want to impose a thought. You are shoving away examples of what you don't want to consider as "unreasonable", whatever that word means in this context. And being "unreasonable" is your only reason for not considering it. And a discussion which got to the point to which a Banach space is topologically "more reasonable" than $\{0\}$ is a discussion I don't want to proceed further. – Aloizio Macedo Jun 12 '16 at 4:02
• @William Looking for an axiomatization of a "continuous structure" is entirely different from wanting to restrict a definition, and a very broad question. Is this what you are looking for?. – Aloizio Macedo Jun 12 '16 at 4:07

If you didn't allow these two sorts of topology (discrete and trivial,) then if $X$ was a space with topology $\tau$, and $Y\subset X$, you couldn't always define a topology on $Y$.

The most obvious example is that the discrete topology on $\mathbb Z$ is the topology you get by considering $\mathbb Z\subset \mathbb R$ with the usual topology on $\mathbb R$.

It's a little harder to get trivial topologies, since you'd have to start with a non-Hausdorff topology.

I'm guessing there are category-theory reasons, also - that limits or co-limits might fail to exist in the category of topological spaces. But that's possibly more advanced than you need, and I'm not sure it is true.

The original definitions of topology had pretty strong axioms, to match metric spaces more closely.

But then people encountered "spaces" where fewer and fewer of the separation rules were satisfied. But I'm not sure I've ever personally seen a topology used in real math that was not $T_0$.

• Comments are not for extended discussion; this conversation has been moved to chat. The conversation started out fine but was becoming a bit too personal towards the end. Anyone interested in reading what was said is instructed to check out the chatroom above. – Jyrki Lahtonen Jun 12 '16 at 5:58
• The "downward" topology on a poset is usually not $T_0$ and has uses in Forcing. – DanielWainfleet Feb 5 '17 at 10:07

This question is way too tl;dr for me, but one phrase that hasn't been thrown around here, as far as I can tell, is "uniform space," and I wonder if that's one way of getting at what you want from a notion of "continuous space," since it is the topological setting were Cauchy completeness makes sense -- one step up in generality from metric spaces. (See also the nLab page.)

• This is a great answer; thank you so much for your help! I guess some (somewhat arbitrary criteria) we might want for a "continuous space" which I'm not sure that uniform spaces satisfy: can we exclude countable spaces or discrete spaces? Do we include almost all real continuous function spaces? Are limits well defined? – Chill2Macht Jun 12 '16 at 14:17

The notion of a topology being a frame of open sets is described by nice algebraic properties and is well behaved under various constructions. This implies that "topological space" is the sort of mathematical structure that is amenable to study.

(In fact, the idea of a frame of open sets is so nice that there has since spawend a theory of locales which keep the frames but discard the notion of points. This is sometimes called "pointless topology")

This means, if whatever particular kind of thing you want to study can be expressed as a topological space, then usually, the most fruitful way to study it is as a "topological space with additional properties".

• cool I've never heard of this before; thank you so much! – Chill2Macht Jun 12 '16 at 6:35

Here are my thoughts on this question, and how I would answer it now:

In what follows, I will use "continuous" in the sense of "continuous enough for an analyst to care about it or find it interesting" and not in the sense of "morphism between topological spaces"; the latter sense is obviously much more general. However, I believe the former sense better captures the layperson's intuition of the word continuous, as well as what Riesz was referring to. Indeed, some consider analysis to consist of "precise formulation of intuitive notions of... continuity", or the study of "continuous change". Only the former sense of the term "continuous" makes sense in these contexts, whereas the latter does not.

Functional analysis is often taught as, and to some extent de facto is, a continuation of real and/or complex analysis. To some extent, the distinction is only relevant for elementary analysis, since problems in the field inevitably lead one to consider function spaces.

What unifies these spaces being studied? That they all are topological vector spaces over $\mathbb{R}$ or $\mathbb{C}$, often with multiple compatible/interesting topologies to choose from. (If we want to include the study of manifolds/differential geometry, then one has to abstract further to "spaces which are locally homeomorphic to topological vector spaces over $\mathbb{R}/\mathbb{C}$.)

One might object that one shouldn't need an "algebraic" structure to define spaces which correspond to the intuitive notion of "continuous", and that topological structure should be all that is necessary. However, I have several responses to this:

1. Remember that the topological notion of continuity doesn't correspond to the intuitive notion conveyed by the use of the word "continuous" in everyday speech.

2. We can think of the algebraic condition of being a vector space over $\mathbb{R}$ or $\mathbb{C}$ as meaning that the space "plays nice" with $\mathbb{R}$ and $\mathbb{C}$. Since these two are in some sense the simplest possible spaces for which we don't have to worry about Cauchy sequences not converging, any space that plays nicely with them will be more likely to correspond to our intuition that a Cauchy sequence "should" converge.

3. If we want to include function spaces, then we need to restrict ourselves to vector spaces, because almost any function space in which we could possibly be interested in (as an analyst) is a vector space.

4. The restriction doesn't matter that much when we generalize to spaces which are "locally homeomorphic to topological vector spaces over $\mathbb{R}/\mathbb{C}$", plus naturally puts in place some of the machinery necessary to have well-defined tangent spaces. And I think tangent spaces also correspond well to our intuitive notion of continuity (which is sometimes called smoothness) - for example, at the point of a cone where there is no well-defined tangent plane, we say that it is "pointy" but not "smooth".

5. Remember that being a topological vector space doesn't mean having a mishmash of topological and algebraic structures -- the two structures have to be compatible with each other, which means to some extent that they have to reinforce each other. So to some extent the purpose of the vector space part is to just impose additional topological conditions.

6. There are many important topological properties which correspond to our intuitive notion of "continuous" which follow from their (topological) vector space structure, I will outline this in further detail below.

7. Empirically speaking, most of the topological counterexamples to the intuitive notion of "continuous space" which I found on $\pi$-base were either not vector spaces (or if they were, they weren't vector spaces over $\mathbb{R}$ or $\mathbb{C}$) or were vector spaces but had a topology which either was not compatible with the vector space structure or was not naturally related to it. This suggested to me that something fundamental was going on here.

The strongest argument, however, for imposing algebraic conditions, is the following:

A group topology is determined by its system of neighborhoods of the identity. In particular, the system of neighborhoods at each point are homeomorphic to one another.

Thus what might seem like an abstract condition actually amounts to an intuitive, almost geometrical one: each point of the space, considered locally, is indistinguishable.

The fact that this isn't true for general topological spaces makes them interesting objects of study; however, it doesn't necessarily correspond to intuitive notions of "continuous".

However, I will add one purely topological condition: a "continuous" space must also be $T1$. This is for at least two reasons:

1. As I have already argued elsewhere, it is most natural for a space which is "continuous" in the intuitive sense to have points which are closed and no points which are open. This should mean in particular that every point in the space has a unique neighborhood system (is distinguishable in the topology), the set of limit points is closed, and that the space does not rely on any single point for its "continuity" (i.e. we still have an open set when removing any individual point). These assumptions make the space much better behaved from a limit convergence standpoint and also inherently remove many examples of "discrete" or "semi-discrete" spaces from consideration.

2. Several very important topological consequences follow only for topological vector spaces which are also $T1$ and it is a safe assumption to make in analysis, hence why many authors (e.g. Rudin) routinely make it.

Let's start deriving the nice properties which will make the crux of my argument.

1. Any such space is not just $T1$, but actually completely regular and has a uniform structure on it. Both of these are also often considered to correspond to the intuitive notion of continuity; we get them for free based on relatively weak axioms.

2. Every such space is not only automatically locally path-connected, but locally arc-connected, since being Hausdorff and locally path-connected imply being locally arc-connected. In other words, it automatically has a bunch of curves (topological 1-manifolds) as subsets, which also corresponds well to our intuitive notion of "continuity".

3. Every such space (except for the trivial one) is uncountable, i.e. has "cardinality of the continuum", because any path-connected space consisting of more than two points is uncountable.

4. Because the space is Hausdorff, every limit is unique! Hence these spaces are particularly good for studying limits. Every compact subset is also closed in these spaces as a result, again in accordance with intuition.

5. This is also already enough structure to define a concept of boundedness: https://en.wikipedia.org/wiki/Bounded_set_(topological_vector_space). While a "continuous space" should be in some sense "deformable", it shouldn't be arbitrarily deformable without changing its "character". Boundedness of sets provides a nice invariant to consider in many contexts.

6. Such a space is discrete if and only if it is trivial, which corresponds beautifully with intuition.

All of this without even imposing a metric! Meaning we still get a lot of good properties but still have enough generality consider non-metrizable spaces of interest in analysis (e.g. weak-* topology).

Another good argument for these spaces corresponding to intuition, however, relates to metric spaces. Specifically, these spaces are metrizable if and only if they have countable neighborhood bases. This provides a very simple and easy criterion determining whether or not a space of this type is metrizable, something which does not hold in general. Moreover, this criterion corresponds very nicely to the intuition afforded by considering "balls of radius 1/n and n" which we can use for local neighborhood bases in Euclidean spaces. Finally, every such space of this type which is metrizable is a Baire space by the Baire Category Theorem.

In short, these spaces have via trial and error naturally singled themselves out as the principle objects of study in analysis and differential topology/geometry, and they have most, if not all, of the properties one would associate with an intuitive notion of "continuous space". They also include all of the standard examples which would come to mind, including topological manifolds and all of the most widely-studied function spaces (according to Wikipedia).

See these other questions of mine for more detail: What is the motivation behind the arbitrary union topological axiom?. Note that neither separability nor second-countability hold for these spaces I mentioned in general, and that first-countability holds if and only if the space is metrizable.

Are there any "default" properties which hold for almost all topological spaces in analysis? (The comments are somewhat useful).

https://math.stackexchange.com/questions/1836475/t-f-properties-of-every-non-trivial-topological-vector-space-over-mathbbr

Corollary of the Birkhoff Kakutani Theorem: first countable topological vector spaces

Can Path Connectedness be Defined without Using the Unit Interval? (The answer to this question leads to the argument that path-connectedness is not too unreasonably strong of a property to ask a space to have.)

Is Every (Non-Trivial) Path Connected Space Uncountable?

Why study non-T1 topological spaces? Basically any non-T1 space is not of interest in analysis, except for some operator algebras, but I don't see any strong motivation to want to include all operator algebras/spectra as belonging to the intuitive notion of "continuous space", even if they are useful for studying them.

UPDATE: On p. 36, chapter 2, of Lee's Introduction to Topological Manifolds, it states that

The most important feature of first countable spaces is that is that they are the spaces in which sequences are sufficient to detect most topological properties.

This statement becomes more accurate in the context of continuous spaces, as I have defined them above, as compared to general topological spaces.

For example, a metric space is compact if and only if it is sequentially compact. A continuous space is metrizable if and only if it is first countable. A first countable space is countably compact if and only if it is sequentially compact. Therefore, countable compactness, sequential compactness, and actual compactness all coincide in a continuous space if and only if it is first countable. This accords better with intuition.

An even more ideal result would be to show that a continuous space is a sequential space if and only if it is first-countable. However, I do not know enough about the latter at this time to confirm or deny this. Nevertheless, the usual direction of implication holds, namely that first countable implies sequential, thus sequential continuity and continuity coincide for all metrizable=first-countable continuous spaces.

It would also be nice to find some relationship between continuous spaces and Frechet-Urysohn spaces: sequential continuity vs. continuity.

Well, let me try to define something which I think should give a reasonable definition of a "continuous space". The idea is to generalize the notion of path-connectedness so it doesn't rely on a mapping to the real numbers.

Here's my definition:

A topological space $X$ is a continuous space if for any pair of points $x_1,x_2\in X$ there exists a connected subset L with $\{x_1,x_2\}\subset L\subset X$ and a dense total order on $L$ so such the order topology on $L$ coincides with its subset topology.

Also note that every path-connected space fits the definition, as the order on the real line induces an order on the path.

• I can give you an example of a non-Hausdorff $T_1$ space that meets this definition, including $L$ being a connected subspace. – DanielWainfleet Feb 5 '17 at 13:41
• @user254665: Then please do so. – celtschk Feb 5 '17 at 13:44
• It also may add other spaces. A connected Souslin line is not path-connected, but meets your def'n. – DanielWainfleet Feb 5 '17 at 13:44
• @user254665: As it generalizes the concept of path-connectedness, it is to be expected that non-path-connected spaces fit the definition. Indeed looking at this Wikipedia page I'd say the connected Suslin line should qualify. – celtschk Feb 5 '17 at 13:51
• @user254665: Actually that was what I intended to add in the next comment. However it turns out my argument had a flaw, and I now indeed think I found a counterexample: Take the product of $\mathbb R$ with the ordinary topology and $\mathbb R$ with the cofinite topology, and then identify all points $(0,x)$. Then for points $x_1$ and $x_2$ the set $L$ can be chosen by going to the zero point "horizontally", and then continuing "horizontally" along the horizontal line through the other point (obviously if both points are of the same "height", you'd just move on that line). – celtschk Feb 5 '17 at 14:44