The locally path-connected coreflection

Say you’ve got some path-connected space $X$ and you want to know about it’s fundamental group $\pi_1(X,x)$. But $X$ isn’t locally path-connected so pretty much any standard tools in algebraic topology aren’t going to help you out. What’s an algebraic topologist to do? This post is about a simple but remarkably useful construction that will give you a locally path-connected spaces $lpc(X)$ which has the same underlying set as $X$ but which does not change the fundamental group (or any homotopy or homology groups).

The construction is based on the following basic fact from general topology: If $X$ is locally path-connected and $U$ is an open set of $X$, then the path-components of $U$ are open in $X$.

Definition 1: Suppose $X$ is a topological space. Let $lpc(X)$ be the space with the same underlying set as $X$ but whose topology is generated by the path-components of the open sets of $X$.

This means a basic open sets in $lpc(X)$ is the path-component $C$ of an open set $U$ in $X$. Let $\mathscr{B}$ be the collection of such basic open sets. The rest of this post will be devoted to exploring the basic properties of this construction.

Some preliminary facts

Proposition 2: $\mathscr{B}$ is actually a basis for a topology on the underlying set of $X$.

Proof. Certainly every point of $x$ is contained is some path-component of $X$. Suppose $U_1$ and $U_2$ are open in $X$ and $x\in U_1\cap U_2$. Let $C_i$ be the path-component of $x$ in $U_i$ for $i=1,2$ and $C$ be the path-component of $x$ in the intersection $U_1\cap U_2$. It suffices to show that $C\subseteq C_1\cap C_2$. If $c\in C$, then there is a path $\gamma$ from $c$ to $x$ in $U_1\cap U_2.$ Since $C_i$ is the path-component of $x$ in $U_i$, the path $\gamma$ must have image in both $C_1$ and $C_2$. Thus $c\in C_1\cap C_2$. $\square$

Proposition 3: The topology of $lpc(X)$ is finer than the topology of $X$. Equivalently, the identity function $id:lpc(X)\to X$ is continuous.

Proof. If $U$ is open in $X$, then $U$ is the union of it’s path-components and is therefore a union of basic open sets in $lpc(X)$. Therefore the topology of $lpc(X)$ is finer than the topology of $X$. $\square$

Here is the most important property of $lpc(X)$.

Theorem 4: Suppose $Y$ is locally path-connected and $f:Y\to X$ is a continuous function. Then the function $f:Y\to lpc(X)$ is also continuous.

Proof. Suppose $C$ is the path-component of an open subset $U$ of $X$ (so that $C$ is a basic open set in $lpc(X)$). Suppose $y\in Y$ such that $f(y)\in C$. Since $f:Y\to X$ is continuous and $U$ is an open neighborhood of $f(y)$ in $X$, there is an open neighborhood $V$ of $y$ in $Y$ such that $f(V)\subseteq U$. Now since $Y$ is locally path-connected, we may find a path-connected open set $W$ in $Y$ such that $y\in W\subseteq V$. It suffices to check that $f(W)\subseteq C$. If $w\in W$, then there is a path $\gamma:[0,1]\to W$ from $y$ to $w$. Now $f\circ\gamma:[0,1]\to f(W)\subseteq f(V)\subseteq U$ is a path from $f(y)$ to $f(w)$. Since $C$ is the path-component of $f(y)$ in $U$, we must have $f(w)\in C$. This proves $f(W)\subseteq C$. $\square$

Another way to think about this is in terms of hom-sets of continuous functions. Here $\mathbf{Top}$ denote the category of topological spaces and continuous functions. Thus $\mathbf{Top}(A,B)$ is the set of all continuous functions $A\to B$.

Corollary 5:  If $Y$ is locally path-connected, then the continuous identity $id:lpc(X)\to X$ induces a bijection $\eta:\mathbf{Top}(Y,lpc(X))\to\mathbf{Top}(Y,X)$ given by composing a map $Y\to lpc(X)$ with $id:lpc(X)\to X$.

Proof. Injectivity of $\eta$ follows from the injectivity of the identity function and surjectivity of $\eta$ follows from Theorem 4. $\square$

Of course, an important case of Theorem 4 is when we take $Y=[0,1]$ to be the unit interval. In this case, the above corollary can be interpreted as the fact that $X$ and $lpc(X)$ have the same paths and homotopies of paths. For one, if $X$ is path-connected, then so is $lpc(X)$.

The original intent was to construct a locally path-connected version of a space $X$ in an “efficient” way. Let’s continue to check that we’ve actually done this.

Theorem 6: $lpc(X)$ is locally path-connected. Moreover, $lpc(X)=X$ if and only if $X$ is locally path-connected.

Proof. Suppose $C$ is a basic open neighborhood of a point $x\in lpc(X)$. By construction of $lpc(X)$, $C$ is the path-component of an open neighborhood in $X$. It is important to notice here that the subspace topologies with respect to the topologies of $X$ and $lpc(X)$ may be different so it is not completely obvious that $C$ is path-connected as a subspace of $lpc(X)$. The above theorem will help us out though. Let $x,y\in C$. Then there is a path $\gamma:[0,1]\to X$ with image in $C$ and $\gamma(0)=x$ and $\gamma(1)=y$. Since $\gamma:[0,1]\to lpc(X)$ is also continuous and has image in the subset $C$, we can conclude that $x$ and $y$ can be connect by a path in the subspace $C$ of $lpc(X)$. Thus $lpc(X)$ is indeed path-connected as a subspace of $lpc(X)$ confirming that $lpc(X)$ is indeed locally path-connected.

For the second statement, it is now clear that if $lpc(X)=X$, then $X$ is locally path-connected. Conversely, if $X$ is locally path-connected, then according to Theorem 4, the continuity of the identity function $X\to X$ implies the continuity of the identity function $X\to lpc(X)$. We already knew the identity $lpc(X)\to X$ was continuous so the topologies of $X$ and $lpc(X)$ must be identical. $\square$

A categorical interpretation

The construction of $lpc(X)$ is a special type of functor called a coreflection function – the idea being that the category $\mathbf{lpcTop}$ of locally path-connected spaces is a subcategory of $\mathbf{Top}$ such that for every object of $\mathbf{Top}$ there is a “most efficient” way to construct a corresponding object of $\mathbf{lpcTop}$.

Definition 7: Suppose $\mathcal{C}$ is a category and $\mathcal{D}$ is a subcategory. We say $\mathcal{D}$ is a coreflective subcategory of $\mathcal{C}$ if the inclusion functor $\mathcal{D}\to\mathcal{C}$ has a right adjoint $R:\mathcal{C}\to\mathcal{D}$ called a coreflection functor.

If we break this definition down, the fact that $R$ is right adjoint to the inclusion means that for every object $c$ of $\mathcal{C}$, there is an object $R(c)$ of $\mathcal{D}$ and a morphism $\eta:c\to R(c)$ in $\mathcal{C}$ which induces a bijection $$\mathcal{C}(d,c)\to \mathcal{D}(d,R(c))\text{ where }f\to \eta\circ f$$ for every object $d$ of $\mathcal{D}$. This is precisely our situation: $R=lpc$ and $\eta=id:lpc(X)\to X$ is the continuous identity.

Theorem 8: $lpc:\mathbf{Top}\to\mathbf{lpcTop}$ is a functor right adjoint to the inclusion functor $\mathbf{lpcTop}\to\mathbf{Top}$.

Proof. We’ve already confirmed that we have all the right ingredients. Let’s just put them together. First, we check that $lpc$ is a functor. We have left to see what it does to morphisms. If $f:X\to Y$ is a continuous function of any spaces, then we may compose it with the continuous identity $lpc(X)\to X$ to get a continuous function $f:lpc(X)\to Y$. Since $lpc(X)$ is locally path-connected, Theorem 4 guarantees that $lpc(f):lpc(X)\to lpc(Y)$ is continuous (notice this is actually the same function, it’s just the spaces have different topologies). Thus $lpc$ is the identity on both underlying sets and functions. From here it is more or less obvious that $lpc$ preserves identities and composition.

Theorem 4 then shows that $lpc$ is in fact right adjoint to the inclusion $\mathbf{lpcTop}\to\mathbf{Top}$ since if $A$ and $B$ are locally path-connected, then $\mathbf{lpcTop}(A,B)=\mathbf{Top}(A,B)$ (i.e. $\mathbf{lpcTop}$ is a full subcategory of $\mathbf{Top}$). The natural bijection $$\mathbf{lpcTop}(Y,lpc(X))\cong \mathbf{Top}(Y,X)$$ also is described in Corollary 5. $\square$

This is why it is appropriate to call $lpc(X)$ the locally path-connected coreflection of $X$ or the $lpc$-coreflection of $X$.

Some Examples

For a real number $r>0$, let $C_r=\{(x,y)|(x-r)^2+y^2=r^2\}$ be the circle of radius $r$ centered at $(r,0)$. Additionally, if $A\subseteq (0,\infty)$, let $\displaystyle C_A=\bigcup_{r\in A}C_r$.

Example: Let $A=\left\{1,...,1+\frac{1}{n},...,1+\frac{1}{3},1+\frac{1}{2},2\right\}$. Then $X=C_A$ is a non-locally path-connected, compact planar set that looks something like this:

This is something like a generalized wedge of circles; in fact $X$ is homeormophic to the reduced suspension of $\{(0,0)\}\cup A$. What is $lpc(X)$? Well the topology should only change near points where $X$ is not locally path-connected. Here that is the set $C_{1}\backslash \{(0,0)\}$. A basic neighborhood $U$ of a point in this set is a union of intervals, which are precisely the the path-components of $U$.

In particular, the arc $C_1\cap U$ is open in $lpc(X)$ illustrating the fact that the circles $C_{1+1/n}$ no longer converge to $C_1$. In particular, $lpc(X)$ is homeomorphic to the following planar set where the “limit” circle is no longer a topological limit.

Since the circles in $lpc(X)$ are “discrete,” the resulting space is a wedge of circles but technically does not have the CW-topology (which would not be first countable). Instead, it has a metrizable topology. To be fair, I kind of doubt that $lpc(Y)$ can always be embedded in $\mathbb{R}^2$ whenever the space $Y$ can. Regardless, we know spaces $X$ and $lpc(X)$ have the same homotopy groups but are not homotopy equivalent (an easy way to prove this is using a topologized version of the fundamental group). In fact, all higher homotopy groups are trivial and both fundamental groups are free on a countably infinite set of generators.

Similarly, if $B=\mathbb{Q}\cap (1,2)$, then $Y=\bigcup_{r\in B}C_r$ is not locally path-connected – it looks like a wedge of circles in which the circles are parameterized by the rationals. But the $lpc$-coreflection $lpc(Y)$ is also a countable wedge of circles (with a metrizable topology) – in fact $lpc(Y)\cong lpc(X)$.

If we take $\mathbb{H}=\bigcup_{n\geq 1}C_{1/n}$, then we get the usual Hawaiian earring space. This is already locally path-connected so $lpc(\mathbb{H})=\mathbb{H}$.

Other examples:

• For any totally path-disconnected space $X$  (i.e. a space in which every path-component is a point) the $lpc$-coreflection $lpc(X)$ must be discrete. So if $\mathbb{Q}$ is the rationals, then $lpc(\mathbb{Q})$ is a countable discrete space. More generally, $[0,1]$ cannot be the countable disjoint union of closed sets so, in general, if $X$ is a countable $T_1$ space, then $lpc(X)$ must be discrete. Similarly, if $C$ is the cantor set, then $lpc(C)$ is an uncountable discrete space.
• One could replace circles in the above example with a similar construction using n-spheres in $\mathbb{R}^{n+1}$ and obtain examples with non-trivial higher homotopy and homology groups.

More algebraic topology

For based spaces $(X,x)$ and $(Y,y)$, let $[(Y,y),(X,x)]$ denote the set of based homotopy classes of based maps $(Y,y)\to (X,x)$.

Theorem 9: If $Y$ is locally path-connected, the identity function $id:lpc(X)\to X$ induces a bijection of homotopy classes $[(Y,y),(lpc(X),x)]\to[(Y,y),(X,x)]$.

Proof. Surjectivity follows directly from Theorem 4. Suppose $f,g:(Y,y)\to(lpc(X),x)$ are maps such that $f,g:(Y,y)\to (X,x)$ are homotopic. Then $Y\times I$ is locally path-connected and the homotopy $H:Y\times [0,1]\to X$ is also continuous with respect to the topology of $lpc(X)$. Thus we obtain a based homotopy $H:Y\times [0,1]\to lpc(X)$ between $f,g:(Y,y)\to(lpc(X),x)$. This shows the function on homotopy classes is injective. $\square$.

In the case that $Y=S^0$ is the two-point space, we see that $lpc(X)\to X$ induces a bijection $\pi_0(lpc(X))\to \pi_0(X)$ of path-components. When $Y=S^n$ is the n-sphere, we get the following corollary.

Corollary 10: The identity function $id:lpc(X)\to X$ induces an isomorphism $\pi_n(lpc(X),x)\to\pi_n(X,x)$ of homotopy groups for all $n\geq 1$ and $x\in X$.

Replacing maps on spheres with maps on the standard n-simplex $\Delta_n$, we see there is a canonical bijection between singular n-chains in $X$ and $lpc(X)$. This means similar arguments give the same result for homology groups.

Corollary 11: The identity function $id:lpc(X)\to X$ induces isomorphisms $H_n(lpc(X))\to H_n(X)$ and $H^n(X)\to H^n(lpc(X))$ of singular homology and cohomology groups for all $n\geq 0$.

One of the limitations of algebraic topology is that most techniques do not apply to non-locally path-connected spaces. For instance, covering spaces of locally path-connected spaces are uniquely determined (up to isomorphism) by the corresponding $\pi_1$ action on the fiber, but this convenience only translates to very special types of non-locally path-connected spaces. As long as the goal is to understand the homotopy and (co)homology groups of the space, and not to characterize the homotopy type, the $lpc$-reflection allows one to assume the space in question is locally path-connected.

Definition 12: A space $X$ is semi-locally simply connected if for every point $x\in X$, there is an open neighborhood $U$ of $x$ such that the inclusion $U\to X$ induces the trivial homomorphism $\pi_1(U,x)\to\pi_1(X,x)$ on fundamental groups.

It’s an important fact from covering space theory that every path-connected, locally path-connected and semi-locally simply connected $X$ admits a universal (simply connected) covering $p:\widetilde{X}\to X$.

Proposition 13: $X$ is semi-locally simply connected if and only if $lpc(X)$ is semi-locally simply connected.

Proof. First suppose $X$ is semi-locally simply connected. Suppose $x\in X$ and $U$ is an open neighborhood $U$ of $x$ such that the inclusion $U\to X$ induces the trivial homomorphism $\pi_1(U,x)\to\pi_1(X,x)$. Let $C$ be the path-component of $x$ in $U$. Then $C$ is an open neighborhood of $x$ in $lpc(X)$. The inclusion $f:C\to X$ induces a homomorphism $j_{\ast}:\pi_1(C,x)\to\pi_1(lpc(X),x)\cong\pi_1(X,x)$ which factors as $\pi_1(C,x)\to\pi_1(U,x)\to\pi_1(X,x)$ where the later homomorphism is trivial. Thus $j_{\ast}$ is trivial.

Conversely, suppose $lpc(X)$ is semi-locally simply connected and $x\in X$. Find an open neighborhood $C$ of $x$ in $lpc(X)$ such that $\pi_1(C,x)\to\pi_1(lpc(X),x)$. We can assume $C$ is a basic neighborhood, so that $C$ is the path-component of an open set $U$ of $X$. If $\alpha:[0,1]\to U$ is a loop based $x$, then it must have image in $C$. Since $\alpha$ is null-homotopic in $lpc(X)$, it must be null-homotopic when viewed as a loop in $X$. Thus $\pi_1(U,x)\to\pi_1(X,x)$ is trivial. $\square$

Corollary 14: If $X$ is path-connected and semi-locally simply connected, then $lpc(X)$ admits a universal covering $p:\widetilde{lpc(X)}\to lpc(X)$.

The composition $q=id\circ p:\widetilde{lpc(X)}\to lpc(X)\to X$ is essentially a universal covering of the space $X$ except it doesn’t exactly satisfy the local triviality part of the definition of a covering map. However, it does have pretty much all of the same lifting properties as a covering map: if $Z$ is path-connected, $\tilde{x}\in\widetilde{lpc(X)}$ locally path-connected, and $f:(Y,y)\to (X,q(\tilde{x}))$ is a map such that $f_{\ast}(\pi_1(Y,y))\subseteq q_{\ast}(\pi_1(\widetilde{lpc(X)},\tilde{x}))$, then there is a unique continuous lift $\tilde{f}:(Z,z)\to(\widetilde{lpc(X)},\tilde{x})$ satisfying $q\circ\tilde{f}=f$.

Take the example of the generalized wedge of circles pictured above. This space does not have a universal covering space but it’s $lpc$-coreflection does. We can conclude that for many non-locally path-connected spaces, there is still a covering theoretic approach to characterizing the structure of the fundamental group – just apply the locally path-connected coreflection first.