Covering Spaces and Lifting

Given: The objects, hypotheses, and choice principles stated above.

A covering map is a continuous surjection $p:E\to B$ such that every $b\in B$ has an open neighbourhood $U$ for which $p^{-1}(U)$ is a disjoint union of open sets $V_j$, called sheets, and each restriction $p{\mathrm{\mid }}_{V_j}:V_j\to U$ is a homeomorphism (def-continuous-map-top, def-homeomorphism-and-open-maps, def-disjoint-union-topology). Such a $U$ is evenly covered, and $p^{-1}(b)$ is the fibre over $b$. A covering is trivial when it is isomorphic over $B$ to a product projection $B\times F\to B$ with $F$ discrete. (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings).

Let $(X,{\mathcal{T}}_X)$ and $(Y,{\mathcal{T}}_Y)$ be topological spaces and let $f:X\to Y$ be a function. Continuity is as in def-continuous-map-top, injections, surjections and bijections as in def-injection-surjection-bijection. $f$ is an open map if $f[U]$ is open in $Y$ for every open $U\subseteq X$, a closed map if $f[F]$ is closed in $Y$ for every closed $F\subseteq X$, and a homeomorphism if $f$ is a continuous bijection whose inverse $f^{-1}:Y\to X$ is also continuous; the spaces are homeomorphic when such an $f$ exists. (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological).

Throughout, a topology is as in def-topological-space, and finite, at most countable and uncountable are as in def-countable, so that "countable" always means "at most countable" and every finite set is countable. Let $X$ be a set. The six families below are topologies on $X$; that each really satisfies (T1), (T2) and (T3) is discharged in full after the list. Among those six is the discrete topology ${\mathcal{T}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}:=\mathcal{P}(X)$, in which every subset is open and hence every subset is also closed. (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies).

Given: The objects, hypotheses, and choice principles stated above.

A covering map is a continuous surjection $p:E\to B$ such that every $b\in B$ has an open neighbourhood $U$ for which $p^{-1}(U)$ is a disjoint union of open sets $V_j$, called sheets, and each restriction $p{\mathrm{\mid }}_{V_j}:V_j\to U$ is a homeomorphism (def-continuous-map-top, def-homeomorphism-and-open-maps, def-disjoint-union-topology). Such a $U$ is evenly covered, and $p^{-1}(b)$ is the fibre over $b$. A covering is trivial when it is isomorphic over $B$ to a product projection $B\times F\to B$ with $F$ discrete. (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings).

Let $(X,\mathcal{T})$ be a topological space (def-topological-space). A separation of $X$ is an ordered pair $(U,V)$ of open, nonempty, disjoint subsets of $X$ with $U\cup V=X$; $X$ is disconnected when a separation of $X$ exists and connected when none does. Since $U$ and $V$ are complementary each is clopen, so a separation is the same thing as a partition of $X$ into two nonempty clopen pieces. A subset $A\subseteq X$ is a connected subset when the subspace $(A,{\mathcal{T}}_A)$ is connected. (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets).

Let $A$ and $B$ be sets (def-injection-surjection-bijection for the terminology). $A$ and $B$ are equinumerous, written $A\approx B$, if there exists a bijection $f:A\to B$; $A$ is dominated by $B$, written $A⪯B$, if there exists an injection $f:A\to B$. (Equinumerous sets, $A\approx B$ and $A⪯B$).

Given: The objects, hypotheses, and choice principles stated above.

Let $p:E\to B$ be a covering and $f:Y\to B$ continuous. A lift of $f$ through $p$ is a continuous map $\tilde{f}:Y\to E$ with $p\circ \tilde{f}=f$. This includes lifts of paths $I\to B$ and of homotopies $Y\times I\to B$; an initial lift prescribes the restriction at time $0$ (def-homotopy-relative-and-path-homotopy, def-path-connected). (Lifts of maps, paths, and homotopies through a covering map).

Let $(X,d)$ be a compact metric space (def-metric-compactness, def-metric-space) and let $\mathcal{U}$ be an open cover of $X$. Then there is a real $\delta >0$, a Lebesgue number for $\mathcal{U}$, such that every nonempty $A\subseteq X$ with $\mathrm{diam}(A)<\delta$ (def-metric-bounded-diameter) satisfies $A\subseteq U$ for some $U\in \mathcal{U}$. Diameters of nonempty subsets of $X$ are defined because a compact space is bounded (thm-compact-subset-is-closed-and-bounded) and a subset of a bounded set is bounded. No choice principle is used. (Every open cover of a compact metric space has a Lebesgue number: a $\delta >0$ such that every nonempty subset of diameter less than $\delta$ lies inside a single member of the cover).

Let $X$, $Y$ and $Z$ be topological spaces, with subspaces carrying the subspace topology (def-subspace-topology-top). Then: 1. Composites. If $f:X\to Y$ and $g:Y\to Z$ are continuous (def-continuous-map-top) then $g\circ f:X\to Z$ is continuous. 2. Open cover. Let $f:X\to Y$ be a function and let $\{U_i:i\in I\}$ be a family of open subsets of $X$ with ${\bigcup }_{i\in I}{U}_i=X$. If $f{\mathrm{\mid }}_{U_i}:U_i\to Y$ is continuous for every $i\in I$, then $f$ is continuous. 3. Finite closed cover. Let $f:X\to Y$ be a function, let $n\ge 1$ and let $F_1,\dots ,F_n$ be closed subsets of $X$ with $F_1\cup \cdots \cup F_n=X$. If $f{\mathrm{\mid }}_{F_k}:F_k\to Y$ is continuous for every $k$, then $f$ is continuous. The converses of claims 2 and 3 hold with no hypothesis on the cover at all: every restriction of a continuous map to a subspace is continuous (def-subspace-topology-top). The finiteness in claim 3 is not removable; see the remarks. (Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous).

Let $(X,\mathcal{T})$ be a topological space (def-topological-space). An open cover of $(X,\mathcal{T})$ is a family $\mathcal{U}\subseteq \mathcal{T}$ of open sets with $X=\bigcup \mathcal{U}$; a subcover of $\mathcal{U}$ is a subfamily that is itself an open cover; and $(X,\mathcal{T})$ is compact when every open cover of it has a finite subcover. (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

Every point of $B$ has an evenly covered open neighbourhood $U$: $p^{-1}(U)$ is a disjoint union of open sheets $V$, and $p{\mathrm{\mid }}_V:V\to U$ is a homeomorphism. (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings).

The closed interval $I=[0,1]$ is compact in the usual metric, and every closed subinterval of $I$ is connected. (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, A subset of $\mathbb{R}$ is connected if and only if it is order-convex, that is, an interval).

For every real $x$ there is an integer $m\ge 1$ with $x<m$. (Every complete ordered field is Archimedean).

Given: The objects, hypotheses, and choice principles stated above.

Let $p:E\to B$ be a covering and $f:Y\to B$ continuous. A lift of $f$ through $p$ is a continuous map $\tilde{f}:Y\to E$ with $p\circ \tilde{f}=f$. This includes lifts of paths $I\to B$ and of homotopies $Y\times I\to B$; an initial lift prescribes the restriction at time $0$ (def-homotopy-relative-and-path-homotopy, def-path-connected). (Lifts of maps, paths, and homotopies through a covering map).

Let $p:E\to B$ be a covering, let $\alpha :I\to B$ be a path, and let $e_0\in E$ satisfy $p(e_0)=\alpha (0)$. There is a unique path $\tilde{\alpha }:I\to E$ with $\tilde{\alpha }(0)=e_0$ and $p\circ \tilde{\alpha }=\alpha$. (Existence and uniqueness of path lifts through a covering map).

The product set. Let $I$ be a set and let $X_i$ be a set for each $i\in I$. The product is ${\prod }_{i\in I}{X}_i:=\{x:x\text{ is a function with domain }I\text{ and }x(i)\in X_i\text{ for every }i\in I\},$ and we write $x_i:=x(i)$, the $i$-th coordinate of $x$. Two elements of the product are equal exactly when they agree at every index, functions being equal when they have the same domain and the same values. For $j\in I$ the $j$-th projection is ${\pi }_j:{\prod }_{i\in I}{X}_i\to X_j,{\pi }_j(x):=x_j\mathrm{.}$. The product topology ${\mathcal{T}}^{\mathrm{\Pi }}$ on ${\prod }_i{X}_i$ is the initial topology of the projections: the topology generated by the subbasis $\{{\pi }_i^{-1}[U]:i\in I,U\in {\mathcal{T}}_i\}$. Finite intersections of subbasic sets form a basis for it, and they are exactly the boxes ${\prod }_{i\in I}{U}_i$ with every $U_i$ open in $X_i$ and $U_i=X_i$ for all but finitely many $i$. (The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space).

Let $X$, $Y$ and $Z$ be topological spaces, with subspaces carrying the subspace topology (def-subspace-topology-top). Then: 1. Composites. If $f:X\to Y$ and $g:Y\to Z$ are continuous (def-continuous-map-top) then $g\circ f:X\to Z$ is continuous. 2. Open cover. Let $f:X\to Y$ be a function and let $\{U_i:i\in I\}$ be a family of open subsets of $X$ with ${\bigcup }_{i\in I}{U}_i=X$. If $f{\mathrm{\mid }}_{U_i}:U_i\to Y$ is continuous for every $i\in I$, then $f$ is continuous. 3. Finite closed cover. Let $f:X\to Y$ be a function, let $n\ge 1$ and let $F_1,\dots ,F_n$ be closed subsets of $X$ with $F_1\cup \cdots \cup F_n=X$. If $f{\mathrm{\mid }}_{F_k}:F_k\to Y$ is continuous for every $k$, then $f$ is continuous. The converses of claims 2 and 3 hold with no hypothesis on the cover at all: every restriction of a continuous map to a subspace is continuous (def-subspace-topology-top). The finiteness in claim 3 is not removable; see the remarks. (Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous).

Let $(X,\mathcal{T})$ be a topological space (def-topological-space). An open cover of $(X,\mathcal{T})$ is a family $\mathcal{U}\subseteq \mathcal{T}$ of open sets with $X=\bigcup \mathcal{U}$; a subcover of $\mathcal{U}$ is a subfamily that is itself an open cover; and $(X,\mathcal{T})$ is compact when every open cover of it has a finite subcover. (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

Given: The objects, hypotheses, and choice principles stated above.

Let $p:E\to B$ be a covering, $H:Y\times I\to B$ a homotopy, and ${\tilde{H}}_0:Y\to E$ a lift of $H(-,0)$. There is a unique lift $\tilde{H}:Y\times I\to E$ of $H$ extending ${\tilde{H}}_0$. (Existence and uniqueness of homotopy lifts through a covering map).

Every covering map is a surjective local homeomorphism, and each of its fibres is discrete in the subspace topology. (Covering maps are surjective local homeomorphisms with discrete fibres).

Let $(X,\mathcal{T})$ be a topological space (def-topological-space). A separation of $X$ is an ordered pair $(U,V)$ of open, nonempty, disjoint subsets of $X$ with $U\cup V=X$; $X$ is disconnected when a separation of $X$ exists and connected when none does. Since $U$ and $V$ are complementary each is clopen, so a separation is the same thing as a partition of $X$ into two nonempty clopen pieces. A subset $A\subseteq X$ is a connected subset when the subspace $(A,{\mathcal{T}}_A)$ is connected. (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets).

Given: The objects, hypotheses, and choice principles stated above.

Let $p:E\to B$ be a covering, $H:Y\times I\to B$ a homotopy, and ${\tilde{H}}_0:Y\to E$ a lift of $H(-,0)$. There is a unique lift $\tilde{H}:Y\times I\to E$ of $H$ extending ${\tilde{H}}_0$. (Existence and uniqueness of homotopy lifts through a covering map).

Let $f:X\to Y$ be continuous and let $x_0\in X$. Composition sends a loop $\alpha$ at $x_0$ to the loop $f\circ \alpha$ at $f(x_0)$. Using the loop classes and fundamental group of def-based-loops-and-fundamental-group, the proposed induced homomorphism is $f_{\ast }:{\pi }_1(X,x_0)⟶{\pi }_1(Y,f(x_0)),f_{\ast }([\alpha ]):=[f\circ \alpha ]\mathrm{.}$ The next theorem proves that this value is independent of the representative, that it is a group homomorphism in the sense of def-group-homomorphism, and that induced maps respect identities, composition and homotopies that fix the basepoint. (The homomorphism on fundamental groups induced by a pointed continuous map).

A topological space $X$ is simply connected when it is nonempty and path-connected (def-path-connected) and, for every $x_0\in X$, the group ${\pi }_1(X,x_0)$ has exactly one element. (Simply connected topological spaces).

Given: The objects, hypotheses, and choice principles stated above.

Let $p:E\to B$ be a covering and $f:Y\to B$ continuous. A lift of $f$ through $p$ is a continuous map $\tilde{f}:Y\to E$ with $p\circ \tilde{f}=f$. This includes lifts of paths $I\to B$ and of homotopies $Y\times I\to B$; an initial lift prescribes the restriction at time $0$ (def-homotopy-relative-and-path-homotopy, def-path-connected). (Lifts of maps, paths, and homotopies through a covering map).

Let $(X,\mathcal{T})$ be a topological space (def-topological-space). A separation of $X$ is an ordered pair $(U,V)$ of open, nonempty, disjoint subsets of $X$ with $U\cup V=X$; $X$ is disconnected when a separation of $X$ exists and connected when none does. Since $U$ and $V$ are complementary each is clopen, so a separation is the same thing as a partition of $X$ into two nonempty clopen pieces. A subset $A\subseteq X$ is a connected subset when the subspace $(A,{\mathcal{T}}_A)$ is connected. (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets).

A covering map is a continuous surjection $p:E\to B$ such that every $b\in B$ has an open neighbourhood $U$ for which $p^{-1}(U)$ is a disjoint union of open sets $V_j$, called sheets, and each restriction $p{\mathrm{\mid }}_{V_j}:V_j\to U$ is a homeomorphism (def-continuous-map-top, def-homeomorphism-and-open-maps, def-disjoint-union-topology). Such a $U$ is evenly covered, and $p^{-1}(b)$ is the fibre over $b$. A covering is trivial when it is isomorphic over $B$ to a product projection $B\times F\to B$ with $F$ discrete. (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings).

Given: The objects, hypotheses, and choice principles stated above.

Let $p:E\to B$ be a covering, let $\alpha :I\to B$ be a path, and let $e_0\in E$ satisfy $p(e_0)=\alpha (0)$. There is a unique path $\tilde{\alpha }:I\to E$ with $\tilde{\alpha }(0)=e_0$ and $p\circ \tilde{\alpha }=\alpha$. (Existence and uniqueness of path lifts through a covering map).

Endpoint-fixed homotopic paths in the base have lifts with the same endpoint whenever their lifts begin at the same point. (The endpoint of a lifted path depends only on its endpoint-fixed homotopy class).

Let $Y$ be connected and let $f,g:Y\to E$ be lifts through the same covering of the same map $Y\to B$. If $f(y_0)=g(y_0)$ for some $y_0\in Y$, then $f=g$. (Two lifts from a connected space that agree at one point agree everywhere).

Let $f:X\to Y$ be continuous and let $x_0\in X$. Composition sends a loop $\alpha$ at $x_0$ to the loop $f\circ \alpha$ at $f(x_0)$. Using the loop classes and fundamental group of def-based-loops-and-fundamental-group, the proposed induced homomorphism is $f_{\ast }:{\pi }_1(X,x_0)⟶{\pi }_1(Y,f(x_0)),f_{\ast }([\alpha ]):=[f\circ \alpha ]\mathrm{.}$ The next theorem proves that this value is independent of the representative, that it is a group homomorphism in the sense of def-group-homomorphism, and that induced maps respect identities, composition and homotopies that fix the basepoint. (The homomorphism on fundamental groups induced by a pointed continuous map).

Let $(X,\mathcal{T})$ be a topological space (def-topological-space) and let $x\in X$. Subsets carry the subspace topology (def-subspace-topology-top); connectedness is def-connected-space and path-connectedness is def-path-connected. $X$ is locally connected at $x$ when for every open $U$ with $x\in U$ there is an open connected $V$ with $x\in V\subseteq U$, and locally connected when this holds at every point; $X$ is locally path-connected at $x$ when for every open $U$ with $x\in U$ there is an open path-connected $V$ with $x\in V\subseteq U$, and locally path-connected when this holds at every point. (Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point).

Let $f:(X,x_0)\to (Y,y_0)$ be continuous with $f(x_0)=y_0$. Then $f_{\ast }([\alpha ])=[f\circ \alpha ]$ is a well-defined group homomorphism, and for pointed continuous maps ${\mathrm{id}}_{\ast }=\mathrm{id}$ and $(g\circ f{)}_{\ast }=g_{\ast }\circ f_{\ast }$ (Induced fundamental-group maps are well defined, functorial and invariant under based homotopy).

Given: The objects, hypotheses, and choice principles stated above.

For a covering $p:E\to B$ and a continuous map $f:X\to B$, define $f^{\ast }E:=\{(x,e)\in X\times E:f(x)=p(e)\}$ with the subspace topology, and let $f^{\ast }p:f^{\ast }E\to X$ be $(x,e)\mapsto x$ (def-product-topology, def-subspace-topology-top). This is the pullback covering space; its covering property is proved in prop-covering-spaces-are-stable-under-restriction-finite-products-and-pullback. (The pullback of a covering space along a continuous map).

A covering map is a continuous surjection $p:E\to B$ such that every $b\in B$ has an open neighbourhood $U$ for which $p^{-1}(U)$ is a disjoint union of open sets $V_j$, called sheets, and each restriction $p{\mathrm{\mid }}_{V_j}:V_j\to U$ is a homeomorphism (def-continuous-map-top, def-homeomorphism-and-open-maps, def-disjoint-union-topology). Such a $U$ is evenly covered, and $p^{-1}(b)$ is the fibre over $b$. A covering is trivial when it is isomorphic over $B$ to a product projection $B\times F\to B$ with $F$ discrete. (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings).

The product set. Let $I$ be a set and let $X_i$ be a set for each $i\in I$. The product is ${\prod }_{i\in I}{X}_i:=\{x:x\text{ is a function with domain }I\text{ and }x(i)\in X_i\text{ for every }i\in I\},$ and we write $x_i:=x(i)$, the $i$-th coordinate of $x$. Two elements of the product are equal exactly when they agree at every index, functions being equal when they have the same domain and the same values. For $j\in I$ the $j$-th projection is ${\pi }_j:{\prod }_{i\in I}{X}_i\to X_j,{\pi }_j(x):=x_j\mathrm{.}$. The product topology ${\mathcal{T}}^{\mathrm{\Pi }}$ on ${\prod }_i{X}_i$ is the initial topology of the projections: the topology generated by the subbasis $\{{\pi }_i^{-1}[U]:i\in I,U\in {\mathcal{T}}_i\}$. Finite intersections of subbasic sets form a basis for it, and they are exactly the boxes ${\prod }_{i\in I}{U}_i$ with every $U_i$ open in $X_i$ and $U_i=X_i$ for all but finitely many $i$. (The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space).

Let $(X,\mathcal{T})$ be a topological space (def-topological-space) and let $S\subseteq X$. The subspace topology (also relative topology) on $S$ is ${\mathcal{T}}_S:=\{U\cap S:U\in \mathcal{T}\},$ the family of traces on $S$ of the open sets of $X$. The pair $(S,{\mathcal{T}}_S)$ is a subspace of $X$. A subset of $S$ that lies in ${\mathcal{T}}_S$ is said to be open in $S$, and relatively open where the ambient space needs emphasis. (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).

Given: The objects, hypotheses, and choice principles stated above.

A covering map is a continuous surjection $p:E\to B$ such that every $b\in B$ has an open neighbourhood $U$ for which $p^{-1}(U)$ is a disjoint union of open sets $V_j$, called sheets, and each restriction $p{\mathrm{\mid }}_{V_j}:V_j\to U$ is a homeomorphism (def-continuous-map-top, def-homeomorphism-and-open-maps, def-disjoint-union-topology). Such a $U$ is evenly covered, and $p^{-1}(b)$ is the fibre over $b$. A covering is trivial when it is isomorphic over $B$ to a product projection $B\times F\to B$ with $F$ discrete. (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings).

Given: The objects, hypotheses, and choice principles stated above.

Fix a covering $p:E\to B$, a basepoint $b_0\in B$, and $e\in p^{-1}(b_0)$. For $[\alpha ]\in {\pi }_1(B,b_0)$, define $e\cdot [\alpha ]$ as the endpoint of the unique lift of $\alpha$ beginning at $e$ (thm-path-lifting-for-covering-maps). Endpoint homotopy invariance makes this well defined (cor-lifted-path-endpoints-depend-only-on-path-homotopy). With the library's traversal-order product this is a right action; the corresponding left action is $[\alpha ]\cdot e:=e\cdot [\alpha {]}^{-1}$ (def-group-action). (The monodromy right action on a covering fibre and its equivalent left-action convention).

For every pointed topological space $(X,x_0)$, the product $[\alpha ][\beta ]=[\alpha \ast \beta ]$ is well defined and makes ${\pi }_1(X,x_0)$ a group. Its identity is the class of the constant loop $c_{x_0}$, and $[\alpha {]}^{-1}=[\bar{\alpha }]$. (Loop classes form the group ${\pi }_1(X,x_0)$ under concatenation).

Throughout, $I:=[0,1]=\{t\in \mathbb{R}:0\le t\le 1\}$ (def-interval) carries the subspace topology inherited from $\mathbb{R}$ with its usual topology (def-subspace-topology-top, lem-real-line-is-a-metric-space, def-metric-topology, def-metrizable-space). It is called the unit interval. A path in $X$ from $x$ to $y$ is a continuous map $\gamma :I\to X$ with $\gamma (0)=x$ and $\gamma (1)=y$, where $I:=[0,1]$ carries the subspace topology inherited from $\mathbb{R}$; $X$ is path-connected when any two of its points are joined by such a path. (Paths, path-connected spaces and path components).

Given: The objects, hypotheses, and choice principles stated above.

For a covering $p:E\to B$, a deck transformation is an isomorphism $h:E\to E$ over $B$, so $p\circ h=p$ (def-map-and-isomorphism-of-covering-spaces). Deck transformations form the deck group $\mathrm{Deck}(p)$ under composition, and this group acts on $E$ by evaluation (def-group, def-group-action). (Deck transformations and the deck-transformation group of a covering).

Let $Y$ be connected and let $f,g:Y\to E$ be lifts through the same covering of the same map $Y\to B$. If $f(y_0)=g(y_0)$ for some $y_0\in Y$, then $f=g$. (Two lifts from a connected space that agree at one point agree everywhere).

A left action of a group $G$ on a set $X$ (def-group-action) is free when $g\cdot x=x⟹g=e$ for every $g\in G$ and $x\in X$. Equivalently, no nonidentity element of $G$ fixes any point of $X$. (A free group action has no nonidentity element fixing a point).

Given: The objects, hypotheses, and choice principles stated above.

A left action of a group $G$ on a space $E$ by homeomorphisms is a covering-space action when every $e\in E$ has an open neighbourhood $U$ such that $gU\cap U=\mathrm{\varnothing }$ for every nonidentity $g\in G$ (def-group-action, def-homeomorphism-and-open-maps). Acting by homeomorphisms means that each map $e\mapsto g\cdot e$ is a homeomorphism of $E$; the underlying set action alone would not make the translates $gU$ open. This condition implies freeness (def-free-group-action) and makes the translates of $U$ pairwise disjoint. (Covering-space actions by disjoint translates of neighbourhoods).

A covering map is a continuous surjection $p:E\to B$ such that every $b\in B$ has an open neighbourhood $U$ for which $p^{-1}(U)$ is a disjoint union of open sets $V_j$, called sheets, and each restriction $p{\mathrm{\mid }}_{V_j}:V_j\to U$ is a homeomorphism (def-continuous-map-top, def-homeomorphism-and-open-maps, def-disjoint-union-topology). Such a $U$ is evenly covered, and $p^{-1}(b)$ is the fibre over $b$. A covering is trivial when it is isomorphic over $B$ to a product projection $B\times F\to B$ with $F$ discrete. (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings).

The quotient topology. Let $(X,\mathcal{T})$ be a topological space (def-topological-space), let $Y$ be a set and let $q:X\to Y$ be a surjection (def-injection-surjection-bijection). The quotient topology on $Y$ induced by $q$ is the final topology of the one-element family $(q)$ (def-initial-and-final-topology): ${\mathcal{T}}_q:=\{V\subseteq Y:q^{-1}[V]\in \mathcal{T}\}\mathrm{.}$ That this is a topology is discharged in def-initial-and-final-topology, where every final topology is verified to satisfy (T1), (T2) and (T3). Dually, $C\subseteq Y$ is closed in ${\mathcal{T}}_q$ exactly when $q^{-1}[C]$ is closed in $X$, because $q^{-1}[Y\setminus V]=X\setminus q^{-1}[V]$. (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection).

For a covering $p:E\to B$, a deck transformation is an isomorphism $h:E\to E$ over $B$, so $p\circ h=p$ (def-map-and-isomorphism-of-covering-spaces). Deck transformations form the deck group $\mathrm{Deck}(p)$ under composition, and this group acts on $E$ by evaluation (def-group, def-group-action). (Deck transformations and the deck-transformation group of a covering).

For a covering with connected total space, two deck transformations agreeing at one point are equal. Consequently the deck group acts freely on the total space. (On a connected covering space, a deck transformation is determined by one point and the deck action is free).

Let $X$ and $Y$ be topological spaces and let $q:X\to Y$ be continuous (def-continuous-map-top). Each of the following three conditions makes $q$ a quotient map (def-quotient-topology). 1. $q$ is a surjection and an open map (def-homeomorphism-and-open-maps). 2. $q$ is a surjection and a closed map. 3. $q$ admits a continuous section: a continuous $s:Y\to X$ with $q\circ s={\mathrm{i}\mathrm{d}}_Y$. (Surjectivity of $q$ is then automatic and need not be assumed.) Neither clause 1 nor clause 2 is necessary: a quotient map need be neither open nor closed. A witness that is a quotient map by clause 3 while failing clauses 1 and 2 is worked on the companion page, and is named in the remarks below. (A continuous open surjection, a continuous closed surjection, and a continuous surjection admitting a continuous section are all quotient maps).

Given: The objects, hypotheses, and choice principles stated above.

A covering map is a continuous surjection $p:E\to B$ such that every $b\in B$ has an open neighbourhood $U$ for which $p^{-1}(U)$ is a disjoint union of open sets $V_j$, called sheets, and each restriction $p{\mathrm{\mid }}_{V_j}:V_j\to U$ is a homeomorphism (def-continuous-map-top, def-homeomorphism-and-open-maps, def-disjoint-union-topology). Such a $U$ is evenly covered, and $p^{-1}(b)$ is the fibre over $b$. A covering is trivial when it is isomorphic over $B$ to a product projection $B\times F\to B$ with $F$ discrete. (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings).