topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
A directed topological space is a topological space $X$ in which there is some ‘sense of direction’. This can happen in various different ways and the level of the ‘directedness’ can be different in different situations, so naturally there are several ‘competing’ ideas, but the beginning of a consensus on what the overarching idea is.
If one bases homotopy theory on the idea of a singular simplex or more generally a singular cell of any shape, then there is no way in which a ‘sense of direction’ can be encode. If we have a path in a space we can go along it (traverse it) in either direction, from 0 to 1 or from 1 to 0. From this perspective a directed space is one in which not every singular cell $\Delta^n \to X$ (for $\Delta^n$ the standard topological simplex) is supposed to be traversable in all directions, in some sense: instead these $k$-dimensional paths may have a direction .
As an example one can base the ‘sense of direction’ on a closed preorder or partial order, (that is a pospace),so that the paths from the directed interval $[0,1]$ with the usual order to the space $X$, can only be ‘traversed’ in one direction. Another example which does not fit into this first type would be the directed circle.
In other words, a circle with direction determined by the anticlockwise sense. Again it is easy to see that there are certain paths that respect the direction, ‘directed paths’ whilst others do not.
So far there exists a well-developed theory for a notion of directed spaces $X$ where 1-dimensional paths given by maps $[0,1] \to X$ from the interval into the space are equipped with a direction. See in particular the book by Marco Grandis on Directed Algebraic Topology listed below. Another suggested notion for modelling directed spaces is that of framed spaces, which is tailored towards certain higher categorical applications.
Note that a directed space is like a generalised space; not every directed space need be a space in the traditional sense, in accordance with the red herring principle. As an instance of this, note that Marco Grandis in his book Directed Algebraic Topology handles the directed homotopy of small categories, and of cubical complexes, since this is useful for comparison an interpretation of directed homotopy ‘invariants’.
Directed spaces are studied in directed homotopy theory, a relatively young topic. In generalization of how a topological space has a fundamental groupoid, a directed space has a fundamental category.
From a homotopy theoretic perspective one would wish that notions of directed spaces might serve to generalize the homotopy hypothesis – which identifies ordinary (undirected) topological spaces with ∞-groupoids, i.e., with (∞,0)-categories – to a more general context where (∞,0)-categories are generalized to (∞,r)-categories with $r \gt 0$:
An (∞,r)-category in this context might correspond to a $r$-directed topological space , one that comes equipped with a notion of orientation of its $k$-cells for $0 \leq k \leq r$, but was impartial on direction above that dimension.
If such a definition exists, it may need to use filtered topological spaces instead of bare topological spaces.
Even in the absence of a homotopy-theoretic definition of $r$-directed space in this sense, from the perspective of homotopy theory one might take the standpoint of the homotopy hypothesis and define a (nice) $r$-directed space to be an (∞,r)-category, just as it makes good sense and is nowadays common practice in algebraic topology to define a nice topological space to be an ∞-groupoid.
See (n,r)-category for more on that.
Urs Schreiber: I haven’t looked at Marco Grandis’ book yet: does it say anything about the homotopy hypothesis in the context of the definition of directed space used there?
Tim Porter: No.
A directed topological space or d-space is pair $(X, d X)$ consisting of a topological space $X$ and a subset $d X \subset C(I,X)$ of continuous maps from the interval $I = [0,1]$ into $X$ – called directed paths or d-paths – satisfying the following conditions:
(constant paths) every constant map $I\to X$ is directed,
(reparametrisation) $dX$ is closed under composition with increasing maps $I\to I$,
(concatenation) $dX$ is closed under path-concatenation: if the d-paths $a, b$ are consecutive in $X$ $(a(1) = b(0))$, then their ordinary concatenation $a+b$ is also a d-path
A morphism of directed topological spaces $f : (X, d X)\to (Y , d Y)$ is a morphism of topological spaces $f: X \to Y$ which preserves directed paths in that for every $\gamma: I \to X$ in $d X$ the path $f_* \gamma : I \stackrel{\gamma}{\to} X \stackrel{f}{\to} Y$ is in $d Y$.
Basic examples of $d$-spaces include:
The standard directed interval is $I_d = ([0,1], d I)$ with $d I$ the set of all monotonic continuous maps $[0,1] \to [0,1]$ is a d-space
Any pospace $X$ gives rise to a d-space by taking the directed paths to be, well, directed paths, i.e. continuous order-preserving maps from $I_d$ to $X$.
Many other example can be found in the references.
A different definition comes from Sanjeevi Krishnan, A Convenient Category of Locally Preordered Spaces, Applied Categorical Structures, 2009, vol. 17, no 5, p. 445-466 (arxiv):
Definition A stream is a tuple $X, \leq_{-}$, where $\leq_{-}$ assigns to each open subset $U \subset X$ a preorder $\leq_U$, such that:
Here, $U_i, i \in I$ is a collection of open sets, and $\bigvee$ is pointwise or of relations.
Remarks
Another way to endow spaces with directions is via framings, i.e. choices of a “basis of the vector space of tangential directions at each point”. Somewhat abstracting this idea, frames may also be thought about in terms of projections, as the following remark motivates.
Let $V$ be an $n$-dimensional vector space with an inner product $g$. The following structures on $V$ are equivalent.
The two structures are related by setting $v_i$ to be the unit vector spanning $\ker(V_i \to V_{i-1})$ such that $V_{i-1} \oplus v_i$ recovers the orientation of $V_i$ (note that all $V_i$ canonically embed in $V$ as the orthogonal complement of the kernel of the composite map $V \to V_i$).
In the absence of inner products, one cannot speak of orthonormal frames any longer. However, sequences of projections can still be defined, and may be regarded as playing the role of “metric-free orthonormal” frames. (A vaguely analogous line of thinking is that a Morse function $M \to \mathbb{R}$ provides useful “direction” information on $M$, e.g. for the construction of handlebodies, that is ultimately independent from any chosen metric on $M$.) Moreover, this approach offers room for generalization by varying the length of the projection sequence and the vector space dimensions.
The standard orthonormal frame of $n$-dimensional euclidean space $\mathbb{R}^n$ consists for the ordered sequence of vectors $e_1 = (1,0,...,0)$, $e_2 = (0,1,0,...,0)$, …, $e_n = (0,...,0,1)$. By the previous remark, this orthonormal frame is equivalently described by the sequence of projections $\pi_i : \mathbb{R}^i = \mathbb{R}^{i-1} \times \mathbb{R} \to \mathbb{R}^{i-1}$ (each $\mathbb{R}^i$ being endowed with standard orientation).
When forgetting basepoints, then the previous remark and example equally apply to affine spaces, now endowing each point in the space with a basis of frames. Using affine standard framed $\mathbb{R}^n$ as our “local models” for framed spaces (or rather compact contractible “patches” of it, that interact nicely with the projections, see below), one may define global framed spaces and their maps.
Inductively in $n \in \mathbb{N}$, an $n$-framed patch $U \subset \mathbb{R}^n$ is a non-empty subspace of $\mathbb{R}^n$ with the property that its projection $\pi_n(U)$ is an $(n-1)$-framed patch, such that $\pi_n : U \to \pi_n(U)$ has fibers of the form $[\gamma_-(u),\gamma_+(u)]$ for two continuous sections $\gamma_\pm : \pi_n(U) \to \pi_n(U) \times \mathbb{R}$. Given two $n$-framed patches $U$ and $V$, a (partial) $n$-framed patch map $F : U \to V$ is a (partial) continuous map that descends along $\pi_n$ to a (partial) $(n-1)$-framed patch map $\pi_n (U) \to \pi_n(V)$.
Note that $n$-framed patches are compact and contractible spaces.
The standard example of an $n$-framed patch is the closed $n$-cube $\mathbf{I}^n = [-1,1]^n \subset \mathbb{R}^n$. However, in general $n$-framed patches need not be “$n$-dimensional”: for instance, the $0$th slice $\{0\} \times \mathbf{I}^{n-1}$ of the $n$-cube is itself an $n$-framed patch, and so are the “$k$-directed intervals” $\mathbf{I}_k := \{0\}^{k-1} \times \mathbf{I} \times \{0\}^{n-k}$.
Let $X$ be a topological space. Fix $n \in \mathbb{N}$.
An $n$-framed chart $(U,\gamma)$ in $X$ is an embedding $\gamma : U \hookrightarrow \mathbb{R}^n$ of a subspace $U \subset X$ whose image $\im(\gamma)$ is an $n$-framed patch.
Two $n$-framed charts $(U,\gamma)$, $(V,\rho)$ in $X$ are compatible if $\rho \circ \gamma^{-1}$ is a partial map of $n$-framed patches.
An $n$-framed space is a space $X$ endowed with an $n$-framing structure $\mathcal{A}$, which is an “atlas” of compatible $n$-framed charts $\{(U_i,\gamma_i)\}$ such that $U_i$ are a locally finite cover of $X$.
The condition for covers to be locally finite is convenient as it describes the situation of locally finite cell complexes. However, the condition could be replaced, and the definition generalized, in several ways. The above version of the definition should be considered a first approximation to a potentially more general notion of framed spaces.
Maps of framed spaces could be defined along the following lines.
Given spaces with $n$-framing structure $(X,\mathcal{A}) \to (Y,\mathcal{B})$ then a framed map $F : X \to Y$ is a map such that for any charts $(U,\gamma) \in \mathcal{A}$ and $(V,\rho) \in \mathcal{B}$, $F$ yields a partial map $\rho \circ F \circ \gamma^{-1} : \mathrm{im}(\gamma) \to \mathrm{im}(\rho)$ of $n$-framed patches.
Framed spaces and framed maps are, in a sense, “very rigid” variants of directed spaces. Nonetheless, they are interesting to study as they turn out to have a rich combinatorial theory associated to them. This combinatorial counterpart is particular useful when translating between the topology of directed spaces and the combinatorics of higher categories: an example of this is the definition of manifold diagrams in the language of framed spaces.
One may want to define directed paths in an $n$-framed space as maps from “the framed interval” into that space: but, in fact, there are now $n$ different framed intervals, corresponding to the $n$ possible directions of $n$-framings. These are precisely modelled by the $n$-patches $\mathbf{I}_k$ mentioned in a previous example. Note, there is a framed bijection $\mathbf{I}_k \to \mathbf{I}_j$ if and only if $k \leq j$ (in particular, directions aren’t interchangeable at all: analogously, note that in an n-category a $(n-k)$-morphisms can be (possibly degenerate) $(n-j)$-morphisms only if $k \leq j$). Given a framed space $(X,\mathcal{A})$, and defining $d_k X = \mathrm{Map}_{\mathrm{fr}}(\mathbf{I}_k, (X,\mathcal{A}))$, we thus obtain a filtration of directed paths:
instead of a single path subset $dX \subset \Map(\mathbf{I},X)$, as required in the definition of $d$-spaces.
But for that to work we need the structure of a directed topological space on $C(I_d,(X,d X))$. This requires that $X$ has directed homotopies! Does Grandis discuss higher directed paths, too? —Urs
Toby: I don't think that you need internal homs and all that. But see my edits to directed object.
Urs: I think we need directed homotopies to check if a “constructed” directed space is actually a directed object in the original definition: that original definition asks us to check if the internal hom $[I,X]$ is weakly equivalent to $X$. Well, I made up this definition because I think it is the right abstraction, but there is room of course to debate this. But if we accept it then we should try to define the internal hom of Grandis’ directed spaces. There is an obvious solution which one should check the details of: namely a directed topological space should be one which singles out not only subsets of $hom(I,X)$ but subsets of $hom(I^{\times n}, X)$ for all $n$, closed under the obvious reparameterization and gluing. This would induce an obvious notion of directed homotopies and should induce in an obvious way an internal hom for directed topological spaces. I’d think. But I don’t feel like investing much time into finalizing this idea right now…
Tim Porter: As I have now looked at Marco’s book, there are results on exponentiable d-spaces.(p.59). I can give details if anyone is still interested.
The above definition of $d$-spaces is from
This has now developed into a book
A discussion of reparameterization of directed paths in directed topological spaces is in
Further references are given in directed homotopy theory.
For framed spaces see:
Christoph Dorn and Christopher Douglas, Framed combinatorial topology, 2021 (pdf)
Christoph Dorn and Christopher Douglas, A brief introduction to framed combinatorial topology, 2022 (pdf)
The above defined directed topological spaces. My impression is that Eric was interested in more general concepts. But the above definition has a straightforward generalization away from topological spaces. The general strategy is really: start with a category with interval object and consider then the category whose objects are pairs $(X, d X)$ for $X$ an object and $d X$ a subobject of $[I,X]$, and whose morphisms are morphisms $X \to Y$ that take $d X$ to $d Y$.
For instance, let’s define directed sets: make the ordinary category Set a category with interval object by , say, taking the interval object to be the set $I := [n]$ of $n$ elements. A map from $I$ into any other set can be regarded as an $n$-step path in that set. Then pairs consisting of a set and a subset of all such maps model “directed sets”.
Eric says: Yes, exactly :) That sounds like a good plan. By the way, what you say about $I := [n]$ reminds me a lot of simplicial sets.
Eric says: We have directed spaces and we may soon have directed sets. This makes me wonder if we should have a directed category internal to another category? This way
Would that make sense?
Urs: Let’s see, before getting into this idea of realizing a directed space as a space internal to something else or the like,
I don’t see what you want to mean by a “directed category”. See, the point is that a category already is supposed to be a combinatorial model for a directed space. Just as a groupoid is a combinatorial model for an undirected space. This is the very motivation for defining directed spaces: to fill in the question marks in
This is why a directed space is defined such that its “thing of all paths in it” is not, in general, a fundamental groupoid but a fundamental category.
Methinks that for the application which you have in mind you want to be studying posets and these are special cases of categories and in particular naturally interpreted as combinatorial models for directed space, in exactly the way in which you are thinking of them as directed spaces! So it seems to me you don’t actually need to be looking for what you seem to be looking for, since it is already quite easily there. But of course maybe I misunderstand what you are after.
Eric: I doubt that what I am looking for is new. If you could help put a name on it, that would be great. I’m not exactly sure what I mean by directed category either other than a “category with a direction” :|
Urs: but a category is directed! Recall that underlying every category is a directed graph (it is a directed graph equipped with a composition operation). So I am still puzzled by what you are looking for, because a “directed category” would have underlying it a “directed directed graph”. What’s that supposed to be? And why do you want it?
Eric: Sorry for being so dense. We can delete this once I get a clue. But for now, I’m still confused. Maybe what I wanted to say is more along the line (but probably still not correct)
“A directed space has a fundamental category”
“A directed set has a fundamental category”
“A directed object has a fundamental category”
Ack! light bulb! (those hurt sometimes)
I think that is probably precisely why you defined directed object.
Could we say (and be correct!) that
“a directed space is a directed object in Top”?
“a directed set is a directed object in Set”?
If so, I think I am making some progress.
Urs: Yes, a directed space should be a directed object in the category of possibly directed topological spaces! (In Top itself there are no directed spaces. Every ordinary topological space is undirected). I think I listed that as a should-be example. To make it a proper example one will have to say a few more probabaly straightforward things about directed homotopies etc. But the idea is certainly this, yes, a directed space is a directed object in the category of possibly directed spaces.
And as for categories: the generic category is a directed object in the category of categories. Unless it happens to be a groupoid. In which case it is an undirected object there.
(All this with respect to the “canonical” choice of interval object. The notion of directedness depends on which interval object you choose to test with. For instance the point itself satisfies the axioms of an interval object. But using it of course everything will look undirected.)
Eric: Ugh. I didn’t want a directed space to be a directed object in the category of directed spaces. That is boring :) A set is an object in the Set too, but it doesn’t tell you anything. Hmm. It looks like what I wanted isn’t going to work as is, i.e. a directed space is not a directed object in Top because there are no directed objects in Top apparently.
Urs: I think you do want that. Just don’t let the terminology let mix you up. An ordinary space is already called a space. While from your perspective an ordinary space ought to be called an undirected space. Then “space” could be assigned to mean “not-necessarily but possibly directed space” and then a directed space could be called a directed object in spaces.
But convention is different. So a directed space is a directed object in the category of “not necessarily but possibly drected spaces”.
Toby: Even here, I don't think that you're really using the terminology ideally. The proper term for what you're calling a “not-necessarily but possibly directed space” is just directed space! Much like a non-associative algebra might happen to be associative, so a directed space might happen to be undirected. (In terms of Grandis's definition, any space $X$ defines a directed space where $d$ consists of only the constant paths.)
Urs: Right, Toby, I think that is my point. I was just trying to convince Eric that there is nothing wrong or cheating or boring about the fact that “a directed space is a directed object in the category of directed spaces”.
But maybe the the true issue is whether we want to speak of “directed objects” over at directed object or rather restrict to speaking about undirected objects. Then every object would be a directed object, possibly with trivial direction information, while those objects which are propertly directed would be the not undirected objects.
I consider you as an authority on such issues of logical rigour. You should say how we should fix the terminology and we’ll implement that.
Toby: I'll discuss this at directed object.
Last revised on September 30, 2022 at 09:00:32. See the history of this page for a list of all contributions to it.