The locally path-connected coreflection (Part I)

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

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 $lpc$ -coreflection 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.

For more, see:

The locally path-connected coreflection (Part II)

The locally path-connected coreflection (Part III)

15 Responses to The locally path-connected coreflection (Part I)

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David Roberts says:

September 8, 2019 at 11:11 am

In the section “A categorical interpretation” there is some broken LaTeX.

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- Jeremy Brazas says:
  
  September 8, 2019 at 12:00 pm
  
  Thanks very much David. I must admit that I don’t see it but would like to fix it if it exists. Would you mind either pointing out exactly what it is or perhaps see if it could have been a browser issue?
  
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  - David Roberts says:
    
    September 8, 2019 at 12:23 pm
    
    I see two pieces of raw input:
    
    $$ \mathcal{C}(d,c)\to \mathcal{D}(d,R(c))\text{ where }f\to \eta\circ f$$
    
    and
    
    $$ \mathbf{lpcTop}(Y,lpc(X))\cong \mathbf{Top}(Y,X)$$
    
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- Jeremy Brazas says:
  
  September 8, 2019 at 12:58 pm
  
  Got it. It looks like I never saw the mistake because my browser was always rendering it. Good to know. Thanks again!
  
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Gabin Kolly says:

December 24, 2020 at 8:27 pm

It’s a nice construction! Thank you for this article.
I wonder if the same construction, but with the connected components instead of the path-connected components, gives a space which is locally connected. It’s easy to see that you can obtain a locally connected space by applying this construction enough times (i.e. by transfinite induction), but perhaps it is sufficient to apply it just one time. Of course, it is not really interesting from the point of view of algebraic topology, I’m just curious.

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- Jeremy Brazas says:
  
  December 28, 2020 at 9:22 pm
  
  Very nice question! I also think the locally connected case is interesting but it’s a bit trickier and it sent me to investigate. Let’s F(X) the underlying set of X with the topology generated by the components of the open sets of X. Now F(X) is the locally connected coreflection of X iff F(X) is locally connected but that doesn’t appear to be obvious at all. I don’t know of a counterexample right now but all references seem to suggest that some exist. However, locally connected spaces are precisely the fixed points of F. To construct the locally connected coreflection lc(X) you need the smallest locally connected topology on X that is larger than the topology of X. Apparently, you need to use F and its fixed points to show that this topology exists! I learned this from Gleason’s very readable paper: https://projecteuclid.org/download/pdf_1/euclid.ijm/1255644959
  
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  - Gabin Kolly says:
    
    December 30, 2020 at 4:53 pm
    
    Thank you for your response!
    The paper is very interesting, it is a neat way of constructing the coreflection.
    Today I found a simple space where applying the construction just once doesn’t suffice to obtain a locally connected space: you can take X to be the open point compactification of the rationals with the euclidean topology, then any open subset which contains the point at infinity is connected, so in F(X) all the points are open except for the point at infinity which has the same open neighborhoods as in X, and F(X) is not locally connected at the point at infinity.
    
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  - Gabin Kolly says:
    
    December 30, 2020 at 4:55 pm
    
    *one point compactification, not open point compactification ^^
    
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