topology (point-set topology, point-free topology)
see also 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 covering space (or wrapping space) is a bundle $p: E \to B$ in Top which is locally trivial and with discrete fiber. That is, a map $p: E \to B$ is a covering space over $B$ if for each point $x \in B$, there exists an open neighborhood $U$ of $x$ evenly covered by $p$: the pullback of $p$ over $U$ is isomorphic to a product bundle with discrete fiber $E_x = p^{-1}(x)$:
(the square is a pullback and the isomorphism maps $(x, e \in E_x) \mapsto e$).
Covering spaces over $B$ form an evident full subcategory $Cov/B \hookrightarrow Top/B$. These can be put together to form a replete, wide subcategory $Cov$ of $Top$, so that $Cov/B$ is an over category, just as the notation suggests.
$\,$
There is a generalizatin to “semi-coverings” (Brazas12). Semicoverings satisfy the “2 out of 3 rule”. I.e,, if $f=g h$ and two of $f,g,h$ are semicoverings , then so is the third. This is not true for covering maps.
Different points in $B$ may have non-isomorphic fibers. However, if open sets $U$ and $V$ are evenly covered by $p: E \to B$ and have nonempty intersection, then there are canonical identifications
between typical fibers over $x \in U$, $y \in V$, and $z \in U \cap V$. If $B$ is path-connected, then all the fibers match up to isomorphism (by the unique path-lifting lemma; see below).
Fibers may be empty. Some authors choose to forbid that, adding the condition that $p$ be a surjection, but the resulting category of covering projections over a space $B$ is not as nice as it would be without that condition.
The terms “covering space” and “covering projection”, while traditional, are certainly not optimal: they mislead by being too close to (open) “coverings”. James Dolan has suggested “wrapping space” as an alternative (as in the image of thread wrapping around a spool, to evoke the archetypal example of a covering projection, $p: \mathbb{R} \to S^1: x \mapsto \exp(i x)$).
Every covering space (even in the more general sense not requiring any connectedness axiom) is an etale space, but not vice versa:
for a covering space the inverse image of some open set in the base $B$ needs to be, by the definition, a disjoint union of homeomorphic open sets in $E$; however the ‘size’ of the neighborhoods over various $e$ in the same stalk required in the definition of étalé space may differ, hence the intersection of their projections does not need to be an open set, if there are infinitely many points in the stalk.
even if the the stalks of the etale space are finite, it need not be locally trivial. For instance the disjoint union $\coprod_i Ui$ of a collecton of open subsets of a space $X$ with the obvious projection $(\coprod_i U_i) \to X$ is etale, but does not have a typical fiber: the fiber over a given point has cardinality the number of open sets $U_i$ that contain this particular point.
The connection between covering spaces over $B$ and the fundamental group $\pi_1(B)$ (for $B$ a connected space) is very old and runs very deep. An updated account involves shifting attention to representations of the fundamental groupoid $\Pi_1(B)$ (regardless of connectedness); we give a brief outline of the theory here.
Under some technical topological assumptions on the space $B$, the fundamental theorem can be stated thus:
(fundamental theorem of covering spaces)
Let $B$ be a topological space which is locally path-connected and semi-locally simply-connected.
Then the category $Cov/B$ of covering spaces over $B$ is equivalent to the category of functors $\Pi_1(B) \to Set$ from the fundamental groupoid of $B$ to Set (permutation representation).
This equivalence is exhibited by a functor
sending a covering space $p: E \to B$ to the functor which maps the object $b \in \Pi_1(B)$ to the fiber $E_b = p^{-1}(b)$. Given a map $[\phi]: b \to c$ in $\Pi_1(B)$, where $\phi$ is a path from $b$ to $c$, the unique path-lifting lemma says that for any $e \in E_b$, there exists a unique fill-in $\tilde{\phi}: I \to E$, that is the diagonal arrow making the following diagram commute:
and we define $Fiber([\phi]): E_b \to E_c$ as the map sending $e \in E_b$ to $\tilde{\phi}(1) \in E_c$.
(e.g Møller 11, theorem 7.8)
In fact, in the proof of this theorem one establishes an adjoint equivalence: one constructs a left adjoint to $Fiber$,
via a tensor product or weighted colimit construction, namely the one that extends (by left Kan extension along the Yoneda embedding on $\Pi_1(B)^{op}$) the functor
that sends each object $b$ of $\Pi_1(B)$ to a universal covering space $\tilde{B}_b$ over the path-component of $b$.
We now spell out the details.
Given a space $B$, let $|B|$ be $B$ retopologized with the discrete topology, and consider the pullback in $Top$
Let $\overline{Path}(B)$ be the quotient of $Path(B)$ by the equivalence relation “homotopy rel boundary”. We can think of $\overline{Path}(B)$ as a sum of spaces
fibered in the obvious way over $|B|$ (the set of all basepoints $b$), where $\tilde{B}_b$ is the space of paths in $B$ which begin at $b$, modulo homotopy-rel-boundary. The space $\tilde{B}_b$ can be thought of the universal covering space over the connected component of a point $b \in B$, considered as a space based at $b$.
We have a span
with an obvious (contravariant) composition action $comp$ of the fundamental groupoid $\Pi_1(B)$, itself regarded as a span
with a monad structure in the bicategory of spans. The action gives a map
of spans from $|B|$ to $B$.
Now suppose given an object $F$ of $Set^{\Pi_1(B)}$, i.e., a covariant action of the fundamental groupoid, that is to say a span $F: 1 \to |B|$ equipped with an action $\alpha$ of the monad $\Pi_1(B): |B| \to |B|$ in $Span(Top)$. The data of a right-handed action $comp$ on $\overline{Path}(B)$ and the left-handed action $\alpha$ on $F$ gives rise to a two-sided bar construction
which here is a simplicial object in the category of spans from $1$ to $B$, whose two face maps from degree 1 to degree 0 take the form:
The coequalizer of this pair provides a canonical augmentation of the two-sided bar construction, and may be called the tensor product
(the seemingly opposite placement of the two tensor factors, as compared against the span constructions above, is simply an artifact of the discrepancy between diagrammatic order of composition, and the traditional order in which right actions are covariant and left actions contravariant).
As a span from $1$ to $B$, that is as a bundle over $B$, this tensor product is indeed a covering space over $B$, assuming that $B$ is locally connected and semi-locally simply connected. Finally, the functor
is under these conditions quasi-inverse to the fiber functor
An abstract way of considering the functor $Fiber$ is that it is obtained by homming:
and this forces its left adjoint to be given by the tensor product construction described above.
As a special case, consider the permutation representation $\Pi_1(B) \to Set$ given by the discrete fibration
David Roberts: shouldn’t such a discrete fibration then give rise to a functor $|B| \to Set$? If you mean $Mor(\Pi_1(B))$, then this could probably be described as the total tangent groupoid, which is the action groupoid for the action of $\Pi_1(B)$ on itself.
Todd Trimble: I didn’t make myself clear then. Recall that if $C$ is an internal category in a category $E$ (with $E = Set$ in this discussion), then one defines $E^C$ by taking its objects to be internal discrete fibrations, defined as arrows $F \to C_0$ equipped with the data of an action by the internal category $C$, considered as a monoid in spans from $C_0$ to $C_0$. (This is a standard usage of the term “discrete fibration”; see Johnstone’s Topos Theory for instance.) Looking over this again, I guess I really should have had $F = Mor(\Pi_1(B))$, and $|B|$ here means the underlying set of $B$. But hopefully my meaning is now clear.
David Roberts: Yes, I see now.
(as a span from $1$ to $|B|$) equipped with the obvious (covariant) action of the monad $\Pi_1(B)$ (as a span from $|B|$ to $|B|$). This is essentially the “regular representation” of the fundamental groupoid. The tensor product of the previous section,
is a way of realizing the universal covering space over $B$.
Here is a way of thinking of this construction which links it to the description of universal bundles by Roberts and Schreiber, which is based on considering tangent spaces of the fundamental groupoid. If the fundamental groupoid $G = \Pi_1(B)$ is connected, its universal bundle (as a fibration of groupoids) may be realized as the “tangent groupoid at $b$” or slice
for a chosen basepoint $b \in B$. Note that this slice groupoid is the pullback
with $I$ the groupoid $(0 \overset{\sim}{\to} 1)$. This is then a groupoid over $G$ by the restriction of $ev_1$.
Since the set of arrows of $G$ is obtained as a quotient of the set of paths in $B$, it inherits naturally a topology (a quotient of the compact-open topology on $B^I$) which, together with the given topology on $G_0 = B$, makes $G$ a topological groupoid. Then we recover the universal covering space $B^{(1)}_b$ (I prefer this notation for the 1-connected cover, rather than the usual $\tilde{B}$, because it generalises to $B^{(n)}$ for $n$-connected covers - DR) over $B$ by pulling back along the functor $B \to G$, where we consider $B$ as a topological groupoid with only identity arrows. The assumptions on the topology of $B$ mean that $G$ is a locally trivial groupoid? with discrete hom-spaces, which implies that $B^{(1)}_b$ is a locally trivial bundle with discrete fibres. Local path-connectedness implies that it is locally trivial, and the local condition on $\pi_1$ holds if and only if the fibres are discrete - this last result is due to Daniel Bliss.
Another way to consider the topological conditions on $B$ is to realise that $\Pi_1(B)$, with its inherited topology, is equivalent to a topologically discrete groupoid (in some appropriate localisation of the 2-category of topological groupoids) if and only if $B$ is locally path-connected and semi-locally simply-connected. Otherwise one has to consider the pro-homotopy 1-type of $B$, as in the theory of algebraic fundamental groups (recall that varieties with appropriate topologies - e.g Zariski - are topologically badly behaved).
David Roberts: Is there a prodiscrete completion of a topological groupoid? Maybe we need to assume it is locally trivial, so it is weakly equivalent (in the said localised 2-category) to a groupoid enriched in $Top$, considered as being internal to $Top$. We could then talk about quotients by wide subgroupoids being topologically discrete. Or even quotients being discrete and having finite Leinster cardinality?? Hmm…
In this analysis, the universal covering space $E_b$ of (path-connected) $B$ is retrieved as the quotient of the space of paths which start at the basepoint $b$, modulo homotopy-rel-boundary; the projection to $B$ takes a class of a path $\phi$ to its terminal point $\phi(1)$. This last description is what one would find in any textbook on algebraic topology dealing with covering spaces. This covering space is, strictly speaking, universal among connected covering spaces
More generally, if $S \subset |B|$ is a set of basepoints (Thanks, Ronnie Brown! - DR), we can form the pullback
which is again a groupoid over $G$ by restriction of $ev_1$. Then pullback of $(S/G) \to G$ along the inclusion $B \to G$ is a covering space which is the sum
of connected, 1-connected covering spaces based at the points in $S$. Thus for not-necessarily-connected $B$, taking $S$ such that it intersects each component of $B$ once we can get a universal covering space of $B$ (universal among covering spaces $E \to B$ that induce isomophisms $\Pi_0(E) \to \Pi_0(B)$).
This construction is functorlal (for general $S\subset |B|$), since a map $(B,S) \to (B',S')$ of pairs (remember we are giving $S,S'$ the discrete topology, not the subspace topology) induces a functor of (topological) groupoids $\Pi_1(B) \to \Pi_1(B')$, which by universality of the pulbacks in the above construction gives a map
covering the given map $B \to B'$.
The dependence on basepoints is of course spurious; we can make this explicit by considering the colimit obtained by pasting together the universal covering spaces $B^{(1)}_b$ along isomorphisms induced by paths $b \to c$. But this is in effect how our tensor product construction of the universal covering space works: $\overline{Path}(B)$ is precisely the sum
which can be viewed as a topological span from $|B|$ to $B$. The fundamental groupoid acts contravariantly on this sum, and the tensor product
is the same thing as the coequalizer of the pair of arrows
in $Top/B$, where one arrow is projection and the other is given by the action of pulling back along classes of paths; this coequalizer is a precise description of the pasting colimit alluded to above. It should be noted that this coequaliser is isomorphic to the covering space $B^{(1)}\langle S\rangle$ when $S$ has one point in each component of $B$, but the description as the tensor product is a priori functorial without reference to a set of basepoints.
David Roberts: I think, though, due to the lifting theorems for covering spaces, that given a map $f:B \to B'$ and basepoint sets $S \subset |B|$, $S' \subset |B'|$ that are not necessarily preserved by $f$, there should be a unique lift of $B^{(1)}\langle S\rangle \to B'$ to $B'^{(1)}\langle S'\rangle$ anyway. This would also make this construction independent, up isomorphism, of the choice of basepoints and probably also functorial.
David Roberts: It won’t be functorial - the lift referred to isn’t unique. The up-to-isomorphism is a non-canonical isomorphism.
(David or Urs: please feel free to sprinkle your own sugar over this, by adapting or even copying what David wrote below based on your paper.)
(David Roberts: unless someone feels the discussion below is essential, it can be deleted.)
David Roberts: My personal favourite way of doing this is to topologise the fundamental groupoid, then form the following strict pullback of topological groupoids
where $b\in B$ is a chosen basepoint and $T_b\Pi(B)$ is the tangent groupoid at the object $b$. This links the ideas that the tangent groupoid is the contractible cover of a groupoid, that the fundamental groupoid is the 1-type of a space and the Whitehead construction of connected covers (pull back the path-fibration along the inclusion of a space into the appropriate Postnikov section).
The topology on the fundamental groupoid can either be constructed with the assumption that $B$ is locally path-connected and semi-locally simply-connected, or be given the quotient topology from the free path space $B^I$. With this inherited topology, the fundamental groupoid is equivalent (in the bicategory of topological groupoids and anafunctors) to the same groupoid considered with the discrete topology if and only if $B$ satisfies the usual conditions for the universal covering space to exist. Thus even when $\Pi_1(B)$ is topologised, it still represents a 1-type for nice $B$. One thing which interests me, even though I have no idea about how to approach it, is how for general $B$ the topologised fundamental groupoid can be considered as a pro-homotopy type, that is, the limit of discrete groupoids, taken in the appropriate (bi)category of topological groupoids.
I would like see several expositions of the construction of the universal covering space, since they illustrate different ideas. They seem tautologously related, but things show a bit more of the differences when one passes to bigroupoids.
The universal covering space is
the source-fibre (at a basepoint) of the topologised fundamental groupoid
the pullback of the tangent groupoid as described above
The pullback of the map $(s,t):Mor(\Pi_1(B)) \to Obj(B)\times Obj(B)$ along the inclusion $\{b\}\times B \to B\times B$
Todd I’ll get back to writing more of what I had planned soon. I haven’t had a chance to digest what you’re writing yet, but I prefer to proceed without having to choose basepoints. I’d like to get you and Urs to have a look though when I get back to this within a few days.
David: Of course - hence the theorem about functors from the fundamental groupoid and not the fundamental group. This is where the full tangent groupoid comes in: it is the pullback
or equivalently the slice $Obj(G)\downarrow id_G$ for an internal groupoid $G$ (internal in $Top$, but extensions to other categories work too). The tangent groupoid at a point $g$ is just the subgroupoid of this gotten by pulling back $TG \to Obj(G)$ along the inclusion $\{g\} \to Obj(G)$. I hadn’t thought about applying this construction to my personal universal covering space recipe, so maybe we need to take the discrete topology on $Obj(G)$. That’s what your pullback square above seems to indicate. Urs’ and my paper [arXiv:0708.1741] has stuff on tangent groupoids for anyone who interested in pitching in.
We want to describe here how the universal covering space of $X$ is the homotopy fiber of the canonical morphism $X \to \Pi_1(X)$, hence the $\Pi_1(X)$-principal bundle classified by this cocycle.
We place ourselves in the context of topological ∞-groupoids and regard both the space $X$ as well as its path ∞-groupoid $\Pi(X)$ and its truncation to the fundamental groupoid $\Pi_1(X)$ as objects in there.
The canonical morphism $X \to \Pi(X)$ hence $X \to \Pi_1(X)$ given by the inclusion of contant paths may be regarded as a cocycle for a $\Pi(X)$-principal ∞-bundle, respectively for a $\Pi_1(X)$-principal bundle.
Let $\pi_0(X)$ be the set of connected components of $X$, regarded as a topological $\infty$-groupoid, and choose any section $\pi_0(X) \to \Pi(X)$ of the projection $\Pi(X) \to \pi_0(X)$.
Then the $\Pi(X)$-principal $\infty$-bundle classified by the cocycle $X \to \Pi(X)$ is its homotopy fiber, i.e. the homotopy pullback
We think of this topological $\infty$-groupoid $UCov(X)$ as the universal covering $\infty$-groupoid of $X$. To break this down, we check that its decategorification gives the ordinary universal covering space:
for this we compute the homotopy pullback
where we assume $X$ to be connected with chosen baspoint $x$ just to shorten the exposition a little. By the laws of homotopy pullbacks in general and homotopy fibers in particular, we may compute this as the ordinary pullback of a weakly equivalent diagram, where the point $*$ is resolved to the universal $\Pi_1(X)$-principal bundle
(More in detail, what we do behind the scenes is this: we regard the diagram as a diagram in the global model structure on simplicial presheaves on Top. In there all our topological groupoids are fibrant, hence all we have to ensure is that one of the morphisms of the diagram becomes a fibration, which is what the passage to $\mathbf{E}_x \Pi_1(X)$ achieves. Then the ordinary pullback in the category of simplicial presheaves is the homotopy pullback in $\infty$-prestacks. Then by left exactness of $\infty$-stackification, the image of that in $\infty$-stacks is still a homotopy pullback. )
The topological groupoid $\mathbf{E}_x \Pi_1(X)$ has as objects homotopy classes rel endpoints of paths in $X$ starting at $x$ and as morphisms $\kappa : \gamma \to \gamma'$ it has commuting triangles
in $\Pi_1(X)$. The topology on this can be deduced from thinking of this as the pullback
in simplicial presheaves on Top. Unwinding what this means we find that the open sets in $Mor(\mathbf{E}_x \Pi_1(X))$ are those where the endpoint evaluation produces an open set in $X$.
Now it is immediate to read off the homotopy pullback as the ordinary pullback
Since $X$ is categorically discrete, this simply produces the space of objects of $\mathbf{E}_x \Pi_1(X)$ over the points of $X$, which is just the space of all paths in $X$ starting at $x$ with the projection $UCov_1(X) \to X$ being endpoint evaluation.
This indeed is then the usual construction of the universal covering space in terms of paths, as described for instance in (Dahlke)
Review includes
Jesper Møller, The fundamental group and covering spaces (2011) (pdf)
Ronnie Brown, chapter 11 of Topology and Groupoids
Karl Dahlke, Covering spaces, Building a universal cover , Math Reference Project
Some of the problems of generalising covering spaces to deal with wild spaces are dealt with in: