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
The following is the classical discussion of universal covering spaces in point-set topology.
(essential uniqueness of simply connected covering spaces)
Let $X$ be a topological space which is
Then if $E_i \overset{p_i}{\to} X$ are two covering spaces over $X$, $i \in \{1,2\}$, which are both path-connected and simply connected, then they are isomorphic as covering spaces.
Since both $E_1$ and $E_2$ are simply connected, the assumption of the lifting theorem for covering spaces is satisfied (this prop.). This says that there are horizontal continuous function making the following diagrams commute:
and that these are unique once we specify the image of a single point, which we may freely do (in the given fiber).
So if we pick any point $x \in X$ and $\hat x_1 \in E_1$ with $p(\hat x) = x$ and $\hat x_2 \in E_2$ with $p(\hat x_2) = x$ and specify that $f(\hat x_1) = \hat x_2$ and $g(\hat x_2) = \hat x_1$ then uniqueness applied to the composites implies $f \circ g = id_{E_{2}}$ and $g \circ f = id_{E_1}$.
(universal covering space)
Let $X$ be a topological space which is
Then a path-connected and simply connected covering space, is called the universal covering space of $X$. This is well-defined, if it exists, up to isomorphism, by prop. 1.
(universal covering space reconstructed from free and transitive fundamental group representation)
Let $X$ be a topological space which is well-connected in that it is
Then a universal covering space of $X$ (def. 1) exists.
By this prop. the covering space is connected and simply connected precisely if its monodromy representation is free and transitive. By the fundamental theorem of covering spaces every permutation representation of the fundamental groupoid $\Pi_1(X)$ arises as the monodromy of some covering space. Hence it remains to see that a free and transitive representation of $\Pi_1(X)$ exists. Let $x \in X$ be any point, then $Hom_{\Pi_1(X)}(x,-)$ is such a representation.
This is a functorial construction of a universal covering spaces
where $Top_*^{wc}$ denotes the full subcategory of pointed topological spaces $Top_*$ on the well-connected spaces and $Cov$ is the subcategory of $Top_*^2$ of pointed maps of spaces with objects the covering space maps.
Specifically, if $X$ is a space with basepoint $x_0$, we define $X^{(1)}$ to be the space whose points are homotopy classes of paths in $X$ starting at $x_0$, with the projection $X^{(1)}\to X$ projecting to the endpoint of a path. We can equip this set $X^{(1)}\to X$ with a topology coming from $X$ so that it becomes a universal covering space as above. As described at covering space, under the correspondence between covering spaces and $\Pi_1(X)$-actions, the space $X^{(1)}$ corresponds to the “regular representation” of $\Pi_1(X)$.
This regular representation can be seen to arise by taking a category of elements in the same way that the regular representation of a group is gotten by taking its action on itself: we can see that the universal covering groupoid $\Pi(X)^{(1)}$ in the slice category $\mathbf{Grpd}/\Pi(X)$ (see the universal covering $\infty$-groupoid below) is just the category of elements of the action of $\Pi(X)$ on itself, and can be topologized in a natural way by lifting the topology on $X$ along the canonical projection $\Pi(X)^{(1)} \to \Pi(X)$; decategorifying this yields $X^{(1)}$.
We describe now how the universal cover construction may be understood from the nPOV, using a fragment of cohesive homotopy theory.
The basic idea is that the universal cover of a space $X$ is the homotopy fiber of the canonical morphism
from $X$ to its fundamental groupoid, which exists if both objects are regarded as “topological ∞-groupoids”. We may think of this as a precise way of the intuitive idea of “forcing $\Pi_1(X)$ to become trivial in a universal way”.
(cohesive higher toposes of topological groupoids)
Let $\mathcal{S}$ be some ∞-cohesive site of spaces and consider
the corresponding cohesive (∞,1)-topos over this site. This is the (∞,1)-category of “topological ∞-groupoids” modeled on $\mathcal{S}$.
A canonical choice relating to the traditional discussion would be $\mathcal{S} = Top^\kappa_{lcont}$ a small full subcategory of Top on locally contractible topological spaces, in which case the objects of $\mathbf{H}$ might be called “locally contractible topological ∞-groupoids”.
A more restrictive choice would be $\mathcal{S} =$ CartSp, in which case the objects of $\mathbf{H}$ might be called Euclidean-topological ∞-groupoids.
In fact for the discussion of just universal covering spaces as opposed to the higher stages in the Whitehead tower it would be sufficient and more natural to take $\mathcal{S}$ the full subcategory of locally simply connected topological spaces and consider just the (2,1)-category $\mathbf{H}$ of stacks over this site.
But the following discussion is completely formal and applies globally to all such realizations. Namely all we need is that
Assume that
$\mathbf{H}$ is a cohesive (n,1)-topos
over the given base (n,1)-topos of n-groupoids.
such that its shape modality preserves homotopy fiber products over discrete objects (objects in the essential image of $\Delta$).
We write
for the induced shape modality and
for its unit morphism ona given object $X$.
Assumption 2 is satisfied for the case of $(\infty,1)$-toposes over an ∞-cohesive site and for $(n,1)$-toposes by an $n$-cohesive site as in example 1: by this prop..
Now consider $X \in \mathbf{H}$ any object, for instance a topological space regarded as a 0-truncated topological infinity-groupoid.
Assume, just for ease of discussion, that $X$ is geometrically connected in that the 0-truncation of its shape is contractible:
Using the axiom of choice in the base topos $\mathbf{H}$ we choose a point
The universal cover of $X$ is the homotopy pullback in
The universal property of $\hat X$ is immediate from the abstract setup:
$\hat X$ is simply connected, in that $\tau_1 ʃ \hat X \simeq \ast$
This is because $ʃ X$ is discrete by construction, and hence so is $\tau_1(ʃ X)$. So by assumpotion 2 applying $ʃ$ to the above square yields another homotopy pullback of the form
Now the long exact sequence of homotopy groups (in $\mathbf{B}$) applied to this homotopy fiber sequence is of the form
which implies that $\pi_1( ʃ X )$ and hence $\tau_1 ʃ$ is indeed trivial.
Let $E \longrightarrow X$ be any other object of $\mathbf{H}_{/X}$ such that $\tau_1(ʃ E) \simeq \ast$, then there is morphism
This is because the naturality of the shape unit and of the truncation unit gives a homotopy commuting square of the form
and thus a cone over the diagram which defines $\hat X$ via its universal property.
This shows that $\hat X$ is the universal cover on abstract grounds.
We may also check explicitly that $\hat X$ is given as the space of homotopy classes of paths in $X$ from the given basepoint. To that end we use that, as least over the site CartSp, the shape of $X$ is represented by the topological path ∞-groupoid. See at shape via cohesive path ∞-groupoid.
The following is old material that deserves to be harmonized a bit more with the above stuff.
Let $X$ be a suitably well behaved pointed space. The universal cover $X^{(1)}$ of $X$ is (equivalent to) the homotopy fiber of $X \to \Pi(X)$ in the (∞,1)-category $\mathbf{H} = Sh_{(\infty,1)}(Top_{cg})$ of topological ∞-groupoids.
In other words, the principal ∞-bundle classified by the cocycle $X \to \Pi_1(X)$ is the universal cover $X^{(1)}$: we have a homotopy pullback square
Urs Schreiber: may need polishing.
We place ourselves in the context of topological ∞-groupoids and regard both the space $X$ as well as its homotopy ∞-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 constant 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
On the other hand, we can view a space $X$ as the little $(\infty,1)$-topos $Sh_{(\infty,1)}(X)$ of $(\infty,1)$-sheaves on $X$. If $X$ is locally connected and locally simply connected in the “coverings” sense, then $Sh(X)$ is locally 1-connected.
In fact, for the construction of the universal cover we require only the (2,1)-topos $Sh_{(2,1)}(X)$ of sheaves (stacks) of groupoids on $X$, so we will work in that context because it is simpler. The construction can be adapted, however, to produce a “universal cover” of any locally 1-connected $(\infty,1)$-topos.
Let $E$ be any (2,1)-topos which is locally 1-connected. This means that in the unique global sections geometric morphism $(E^*,E_*)\colon E\to Gpd$, the functor $E^*$ has a left adjoint $E_!\colon E \to Gpd$, which is automatically $Gpd$-indexed. The fundamental groupoid of $E$ is defined to be $\Pi_1(E)\coloneqq E_!(*)$, where $*$ is the terminal object of $E$.
As discussed here in the $(\infty,1)$-case, the construction of $\Pi_1(E)$ is a left adjoint to the inclusion of groupoids into locally 1-connected (2,1)-toposes (which sends $G\mapsto Gpd/G \simeq Gpd^G$). Thus we have a geometric morphism $E\to \Pi_1(E)$ (where we regard $\Pi_1(E)$ as the (2,1)-topos $Gpd^{\Pi_1(E)}$).
Suppose, for simplicity, that $E$ is connected. Then $\Pi_1(E)$ is also connected, and so we have an essentially unique functor $*\to \Pi_1(E)$. We define the universal cover of $E$ to be the pullback (2,1)-topos:
(where of course $*$ denotes the terminal (2,1)-topos $Gpd$).
Now observe that $*\to \Pi_1(E)$ is a local homeomorphism of toposes, since we have $Gpd \simeq Gpd/\Pi_1(E)/(*\to \Pi_1(E))$. Since local homeomorphisms of toposes are stable under pullback, $E^{(1)}\to E$ is also a local homeomorphism, i.e. there exists an object $\widetilde{E}\in E$ and an equivalence $E^{(1)} \simeq E/\widetilde{E}$ over $E$. Moreover, it is not hard to see that $\widetilde{E}$ can be identified with the pullback
in $E$. Note that the bottom map $*\to E^*(E_!(*))$ is $E^*$ applied to the unique map $*\to E_!(*)$, while the right-hand map is the unit of the adjunction $E_!\dashv E^*$.
In order to see that this is a sensible definition, we first observe that $E^{(1)}$ is itself locally 1-connected (since it is etale over $E$). Moreover, it is actually 1-connected, which is equivalent to saying that $E_!(\widetilde{E}) = *$. This is because the “Frobenius reciprocity” condition for the adjunction $E_!\dashv E^*$ (which is equivalent to saying that $E_!$ is $Gpd$-indexed) applied to the defining pullback of $\widetilde{E}$ implies that we also have a pullback
which clearly implies that $E_!(\widetilde{E}) = *$.
Thus, $E^{(1)}$ is a connected and simply connected space with a local homeomorphism to $E$, but is it a covering space? In other words, is it locally trivial? Since we have supposed that $E$ is locally 1-connected, as a (2,1)-category it can be generated by 1-connected objects, i.e. objects $U$ such that $E_!(U)\simeq *$. In particular, we have a 1-connected object $U$ and a regular 1-epic? $U\to *$.
We claim that if $U$ is any 1-connected object of $E$, then $\widetilde{E}$ is trivialized (or split) over $U$, in that $U\times \widetilde{E}$ is equivalent, over $U$, to $U\times E^*S$ for some $S\in Gpd$. For pulling back the defining pullback to $U$, we obtain
But $U\times E^*(\Pi_1(E)) \cong (E/U)^* (\Pi_1(E))$, so to give a map $U \to (E/U)^* (\Pi_1(E))$ over $U$ is the same as to give a map $(E/U)_!(*) \to \Pi_1(E)$ in $Gpd$. But $(E/U)_!(*)\simeq *$, since $U$ is 1-connected, and $\Pi_1(E)$ is connected, so there is only one such morphism. Therefore, the two maps $U\to U\times E^*(\Pi_1(E))$ in the pullback above are in fact the same, and in particular both are the pullback to $E/U$ of the map $*\to \Pi_1(E)$. Thus, $U\times \widetilde{E}$ is equivalent to $(E/U)^*(S) \cong U\times E^*(S)$, where $S= \Omega(\Pi_1(E))$ is the loop object of $\Pi_1(E)$, i.e. what we might call the fundamental group of the connected (2,1)-topos $E$.
Therefore, since $\widetilde{E}$ is trivialized over any 1-connected object, and $E$ is generated by 1-connected objects, $\widetilde{E}$ is locally trivial. Moreover, since $*$ is a discrete object of $E$, so is $\widetilde{E}$. Thus, if we specialize all this to the case $E=Sh_{(2,1)}(X)$ of (2,1)-sheaves on a topological space, then we conclude that $\widetilde{E}$ is an honest 1-sheaf on $X$ which, when regarded as a local homeomorphism over $X$, is locally trivial (hence a covering space), connected, and 1-connected—i.e. a universal cover of $X$.
(real line is universal covering of circle)
Let
$\mathbb{R}^1$ be the real line with its Euclidean metric topology;
$S^1 \coloneqq \left\{ x\in \mathbb{R}^2 \;\vert\; {\Vert x\Vert} = 1 \right\} \subset \mathbb{R}^2$ be the circle with its subspace topology induced from the Euclidean plane.
Consider the continuous function
This exhibits the universal covering space (def. 1) of the circle.
Let $p \in S^1$ be any point. It is clear that we have a homeomorphism of the form
and hence a homeomorphism of the form
Now for $p_1 \neq p_2$ two distinct point in $S^1$, their complements constitute an open cover
and so this exhibits $p \colon \mathbb{R}^1 \to S^1$ as being covering spaces.
Now
$S^1$ is path-connected and locally path connected (this example);
$\mathbb{R}^1$ is simply connected (this example).
Therefore $p$ exhibits $\mathbb{R}^1$ as a universal covering space of $S^1$, by def. 1.
See the references at covering space.