Covering Spaces and Lifting — Examples

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).

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).

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.

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).

For a covering $p:E\to B$, the cardinality of $p^{-1}(b)$ is locally constant as a function of $b\in B$. If $B$ is connected, all fibres are equinumerous. (The cardinality of a covering fibre is locally constant and is constant on a connected base).

The underlying set. Let $I$ be a set and let $X_i$ be a set for each $i\in I$. The disjoint union is ${⨆}_{i\in I}{X}_i:={\bigcup }_{i\in I}(X_i\times \{i\}),$ whose elements are the pairs $(x,i)$ with $i\in I$ and $x\in X_i$. For $j\in I$ the $j$-th canonical injection is ${\kappa }_j:X_j\to {⨆}_{i\in I}{X}_i,{\kappa }_j(x):=(x,j)\mathrm{.}$ The construction is what makes the word "disjoint" honest. Each ${\kappa }_j$ is injective (def-injection-surjection-bijection), since $(x,j)=(x^{\mathrm{\prime }},j)$ forces $x=x^{\mathrm{\prime }}$; the images ${\kappa }_j[X_j]=X_j\times \{j\}$ are pairwise disjoint, since the second coordinate determines $j$; and their union is the whole set. So no assumption that the $X_i$ are disjoint as sets is needed, and none is made: the tag $i$ separates the copies even when $X_i=X_{i^{\mathrm{\prime }}}$ for $i\mathrm{\ne }{i}^{\mathrm{\prime }}$. (The disjoint union (coproduct) ${⨆}_i{X}_i$ with the final topology of the canonical injections: a set is open exactly when each of its traces is).

Throughout, $\mathbb{R}$ is the complete ordered field (def-complete-ordered-field, def-ordered-field) with its order (def-real-order). A subset $I\subseteq \mathbb{R}$ is order-convex when $x,y\in I$ and $x\le z\le y$ imply $z\in I$, and the intervals of $\mathbb{R}$ are the nine listed forms, among them $[a,b]=\{x:a\le x\le b\}$ and $(a,b)=\{x:a<x<b\}$. (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length).

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

For a covering $p:E\to B$, the cardinality of $p^{-1}(b)$ is locally constant as a function of $b\in B$. If $B$ is connected, all fibres are equinumerous. (The cardinality of a covering fibre is locally constant and is constant on a connected base).

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.

For a covering-space action of $G$ on $E$, the orbit map $E\to E\mathrm{/}G$ is a covering. If $E$ is path-connected, the deck group of this covering consists exactly of the transformations supplied by $G$. (The orbit map of a covering-space action is a covering, with the acting group equal to the deck group when the total space is path-connected).

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).

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).

On the set $\mathbb{N}\times \mathbb{N}$ of pairs of natural numbers, define $(a,b)\sim (c,d)⟺a+d=b+c\mathrm{.}$ This is an equivalence relation (lem-int-equivalence). The integers are the quotient $\mathbb{Z}:=(\mathbb{N}\times \mathbb{N})\mathrm{/}\sim ,$ and we write $[(a,b)]$ for the equivalence class of $(a,b)$. (The integers as equivalence classes of pairs of naturals).

Identify $\mathbb{Z}$ with its canonical copy inside $\mathbb{R}$ along the embeddings $\mathbb{N}\to \mathbb{Z}\to \mathbb{Q}\to \mathbb{R}$; then for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$, written $\lfloor x\rfloor$ (Integer part: for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$).

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

For the quotient by integer translation, $q:\mathbb{R}\to \mathbb{R}\mathrm{/}\mathbb{Z}$ is a covering map, and every deck transformation is a unique translation $x\mapsto x+n$ with $n\in \mathbb{Z}$. (The quotient $\mathbb{R}\to \mathbb{R}\mathrm{/}\mathbb{Z}$ is a covering with integer translations as deck transformations).

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).

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).

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

For the quotient by integer translation, $q:\mathbb{R}\to \mathbb{R}\mathrm{/}\mathbb{Z}$ is a covering map, and every deck transformation is a unique translation $x\mapsto x+n$ with $n\in \mathbb{Z}$. (The quotient $\mathbb{R}\to \mathbb{R}\mathrm{/}\mathbb{Z}$ is a covering with integer translations as deck transformations).

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).

For a covering $p:E\to B$, the cardinality of $p^{-1}(b)$ is locally constant as a function of $b\in B$. If $B$ is connected, all fibres are equinumerous. (The cardinality of a covering fibre is locally constant and is constant on a connected base).

On the set $\mathbb{N}\times \mathbb{N}$ of pairs of natural numbers, define $(a,b)\sim (c,d)⟺a+d=b+c\mathrm{.}$ This is an equivalence relation (lem-int-equivalence). The integers are the quotient $\mathbb{Z}:=(\mathbb{N}\times \mathbb{N})\mathrm{/}\sim ,$ and we write $[(a,b)]$ for the equivalence class of $(a,b)$. (The integers as equivalence classes of pairs of naturals).

Let $a,b\in \mathbb{Z}$ with $b>0$. Then there exist integers $q$ and $r$ with $a=qb+r$ and $0\le r<b$, and this pair is unique (Division with remainder in $\mathbb{Z}$: for $a\in \mathbb{Z}$ and $b>0$ there are unique $q,r\in \mathbb{Z}$ with $a=qb+r$ and $0\le r<b$).

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).

If $X$ is any space and $F$ is a nonempty discrete space, the projection $X\times F\to X$ is a trivial covering with fibre $F$. If $X=\mathrm{\varnothing }$, the same holds for $F=\mathrm{\varnothing }$; for nonempty $X$, an empty fibre would violate surjectivity. (Trivial coverings are products with a discrete fibre).

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

The loop $t\mapsto [t]$ in $\mathbb{R}\mathrm{/}\mathbb{Z}$ is not nullhomotopic. (The projected unit interval is not nullhomotopic in $\mathbb{R}\mathrm{/}\mathbb{Z}$).

If a space admits a universal covering, then it is semilocally simply connected. No local path-connectedness hypothesis is required. (A space admitting a universal covering is semilocally simply connected).

A space $X$ is semilocally simply connected at $x\in X$ when there is a neighbourhood $U$ of $x$ and a basepoint-preserving inclusion $(U,x)\hookrightarrow (X,x)$ whose induced map on fundamental groups is trivial (def-neighbourhood-top, def-induced-homomorphism-on-fundamental-groups, def-based-loops-and-fundamental-group). It is semilocally simply connected when this holds at every point. The neighbourhood need not itself be simply connected. (Semilocally simply connected spaces with explicit basepoint convention).

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).