Wild Topology

The locally path-connected coreflection (Part III)

Posted on December 6, 2014 by Jeremy Brazas

This post gives another look at the locally path-connected coreflection that I’ve found quite interesting and useful. For the previous posts see Part I and Part II. In this post, we’ll be motivated by the following two basic observations.

Lemma 1: The topological sum $\coprod_{j\in J}X_j$ of a family of locally path-connected spaces $X_j$ is locally path-connected.

Lemma 2: The quotient of a locally path-connected space is locally path-connected.

Proof. Let $q:X\to Y$ be a quotient map where $X$ is locally path-connected. Let $U$ be an open set in $Y$ and $C$ be a non-empty path-component of $U$ . It suffices to check that $C$ is open in $Y$ . Since $q$ is quotient, we need to check that $q^{-1}(C)$ is open in $X$ . If $x\in q^{-1}(C)\subseteq q^{-1}(U)$ , then there is an open, path connected neighborhood $V$ such that $x\in V\subseteq q^{-1}(U)$ . We claim that $V\subseteq q^{-1}(C)$ . Let $v\in V$ and $\alpha:[0,1]\to V$ be a path from $x$ to $v$ . Then $q\circ\alpha:[0,1]\to q(V)\subseteq U$ is a path from $q(x)\in C$ to $q(v)$ . Since $C$ is the path component of $q(x)$ in $U$ , the path $q\circ\alpha$ must have image entirely in $C$ . Thus $\alpha$ has image in $q^{-1}(C)$ . In particular, $\alpha(1)=v\in q^{-1}(C)$ . $\square$

$\mathbf{Top}$ denotes the usual category of topological spaces and continuous functions. Here is an important definition from Categorical Topology.

Definition 3: Let $\mathscr{C}$ be a subcategory of $\mathbf{Top}$ . The coreflective hull of $\mathscr{C}$ is the full subcategory $H(\mathscr{C})$ of $\mathbf{Top}$ consisting of all spaces which are the quotient of a topological sum (i.e. disjoint union) of spaces in $\mathscr{C}$ .

Certainly $\mathscr{C}\subseteq H(\mathscr{C})$ . Moreover, the coreflective hull is the category “generated” by the collection of spaces $\mathscr{C}$ in the following sense: the inclusion $H(\mathscr{C})\to\mathbf{Top}$ has a right adjoint (called a coreflection) $c:\mathbf{Top}\to H(\mathscr{C})$ . In particular, $c(X)$ is the space with the same underlying set as $X$ but a set $U\subset c(X)$ is open if and only if $f^{-1}(U)$ is open in $Z$ for every map $f:Z\to X$ where $Z\in\mathscr{C}$ . In short, $c(X)$ has the final topology with respect to the collection of all maps from spaces in $\mathscr{C}$ to $X$ .

In the case when $\mathscr{C}=\mathbf{lpcTop}$ is the category of locally path-connected spaces, it follows from Lemmas 1 and 2 that $\mathbf{lpcTop}=H(\mathbf{lpcTop})$ is it’s own coreflective hull. Thus the locally path-connected coreflection $lpc:\mathbf{Top}\to\mathbf{lpcTop}$ discussed in depth in previous posts (Part I and Part II) is precisely the right adjoint. The reason for singling out $\mathbf{lpcTop}$ among other coreflective subcategories of spaces is that (1) local path-connectivity is important in (particularly wild) algebraic topology and (2) the functor $lpc$ has a remarkably simple description: the topology of $lpc(X)$ is generated by the path-components of the open sets of $X$ .

It turns out there is an even simpler way to generate $\mathbf{lpcTop}$ as a coreflective subcategory. I first learned about the following general construction from some fantastic unpublished notes [1] of Jerzy Dydak.

Definition 4: Let $(J,\leq )$ be a directed set. The directed wedge of a collection of spaces $\{(X_j,x_j)|j\in J\}$ indexed by $J$ is the wedge sum $X=\displaystyle\widetilde{\bigvee}_{j\in J}X_j$ (given by identifying the basepoints $x_j$ to a single point $b_0$ ) with the following topology: A set $U\subseteq X$ is open if and only if

$U\cap X_j$ is open in $X_j$ for every $j\in J$ .
if $b_0\in U$ , then there is a $k$ such that $X_j\subset U$ for all $j\geq k$ .

In particular if $(X_j,x_j)=([0,1],0)$ is the unit interval for every $j\in J$ , then $ah(J)=\widetilde{\bigvee}_{j\in J}[0,1]_j$ is the $J$ -arc-hedgehog.

Example 5: If $\omega=\{1,2,...\}$ is the natural numbers, then the $\omega$ -arc hedgehog space is the space $ah(\omega)$ of a sequence of shrinking intervals joined at a point.

Omega arc-hedgehog

In the case that $J=\omega$ and $X_j=S^1$ is the unit circle. The directed wedge $\widetilde{\bigvee}_{\omega}S^1$ is the earring space.

Lemma 6: If $\{X_j|j\in J\}$ is a collection of path-connected and locally path-connected spaces, then $\widetilde{\bigvee}_{j\in J}X_j$ is path-connected and locally path-connected.

In particular, arc-hedgehogs are path-connected and locally-path connected.

Theorem 7: Let $\mathscr{A}$ be the subcategory of all $J$ -arc hedgehogs. Then $H(\mathscr{A})=\mathbf{lpcTop}$ .

Proof. Since every arc-hedgehog is locally path-connected, have $\mathscr{A}\subseteq\mathbf{lpcTop}$ and thus $H(\mathscr{A})\subseteq H(\mathbf{lpcTop})=\mathbf{lpcTop}$ . For the other inclusion, suppose $X$ is locally path-connected. Suppose $U\subseteq X$ . Clearly if $U$ is open, then $f^{-1}(U)$ is open in $ah(J)$ for every map $f:ah(J)\to X$ . For the converse, suppose $U$ is not open. There exists a point $x\in U$ such that for every path-connected neighborhood $V$ of $x$ , there is a point $z_V\in V\backslash U$ . Let $J$ be the directed set of path-connected neighborhoods $V$ of $x$ . For each $V\in J$ , find a path $\alpha_V:[0,1]\to V$ from $x$ to $z_V$ . Define a map $f:ah(J)\to X$ so that the restriction to the $V$ -th arc is the path $\alpha_{V}:[0,1]_{V}\to V$ . It is easy to see that $f$ is continuous based on how we defined the topology of $ah(J)$ . Since $f(b_0)=x$ , we have $b_0\in f^{-1}(U)$ , however, if $1_V$ denotes the end of the $V$ -th arc $[0,1]_V$ , then $f(1_V)=z_V\notin U$ . Thus $1_V\notin f^{-1}(U)$ for all $V\in J$ . It follows that $f^{-1}(U)$ cannot be open since $1_V\to b_0$ is a net in $ah(J)$ converging to the joining point $b_0$ . $\square$

Thus the topology of a locally path-connected space $X$ is entirely determined by maps from arc-hedgehog spaces. Notice that if $X$ is first countable, then we only need to use the $\omega$ -arc hedgehog $ah(\omega)$ .

References:

[1] Dydak, J. Coverings and fundamental groups: a new approach. Preprint. arXiv:1108.3253

Posted in Categorical Topology, Category Theory, coreflection functor, locally path connected | Tagged arc hedgehog, categorical topology, coreflective hull, locally path connected space, quotient space | 8 Comments

The locally path-connected coreflection (Part II)

Posted on November 6, 2014 by Jeremy Brazas

In the last post, I discussed how to efficiently change the topology of a space $X$ in order to obtain a locally path-connected space $lpc(X)$ without changing the homotopy or (co)homology groups of the space in question. This is a handy thing to have hanging from your tool belt but there are some reasonable concerns that come along with this type of construction.

Say $X$ is metrizable. Must the $lpc$ -coreflection also be metrizable? It turns out the answer is yes but that we might lose separability along the way. In this post, we’ll walk through the details which I learned from some unpublished notes of Greg Conner and David Fearnley.

Theorem: If $X$ is path-connected and metrizable, then there is a metric inducing the topology of $lpc(X)$ such that the identity function $id:lpc(X)\to X$ is distance non-increasing.

Proof. Suppose $X$ is a space whose topology is induced by a metric $d$ . Define a distance function $\rho$ on $lpc(X)$ as follows: For any path $\alpha:[0,1]\to X$ and $t\in[0,1]$ , let $\ell_t(\alpha)=d(\alpha(0),\alpha(t))+d(\alpha(t),\alpha(1)).$

Observe that $d(\alpha(0),\alpha(1))\leq\ell_t(\alpha)$ for any $t$ by the triangle inequality.

Now let $\ell(\alpha)=\sup\{\ell_t(\alpha)|t\in[0,1]\}$ .

For points $a,b\in X$ , we define our metric as $\rho(a,b)=\inf\{\ell(\alpha)|\alpha\text{ is a path from }a\text{ to }b\}$

Since $d(a,b)\leq\ell(\alpha)$ for any path $\alpha$ from $a$ to $b$ , we get that $d(a,b)\leq \rho(a,b)$ showing that the identity $lpc(X)\to X$ is non-increasing.

We should still check that $\rho$ is actually a metric which induces the topology of $lpc(X)$ .

Some notation first: If $\alpha,\beta$ are paths in $X$ such that $\alpha(1)=\beta(0)$ , then $\alpha^{-}(t)=\alpha(1-t)$ denotes the reverse of $\alpha$ and $\alpha\cdot\beta$ denotes the usual concatenation of paths

$\alpha\cdot\beta(t)=\begin{cases} \alpha(2t) & 0\leq t\leq 1/2\\ \beta(2t-1) & 1/2\leq t\leq 1 \end{cases}$ .

Notice that $\ell(\alpha)=\ell(\alpha^{-})$ and given $t\in[0,1/2]$ , we have

and if $t\in[1/2,1]$ , we have

Thus, in general, $\ell(\alpha\cdot\beta)\leq\ell(\alpha)+\ell(\beta)$ . Now we can check that $\rho$ is a metric.

If $a=b$ , then we may take $\alpha$ to be the constant path at this point. Then $\ell(\alpha)=0$ showing $\rho(a,b)=0$ . Conversely, if $a\neq b$ , consider any path $\alpha:[0,1]\to X$ from $a$ to $b$ . Find $0<t<1$ such that $\alpha(t)\notin\{a,b\}$ . Then $0<d(a,b)\leq \ell_t(\alpha)\leq\ell(\alpha)$ . Since $\alpha$ was arbitrary, we have $\rho(a,b)>0$ .
Symmetry $\rho(a,b)=\rho(b,a)$ is clear since for every path $\alpha$ from $a$ to $b$ , there is a unique reverse path $\alpha^{-}$ from $b$ to $a$ with $\ell(\alpha)=\ell(\alpha^{-})$ .
Suppose $a,b,c\in X$ . Let $\alpha$ be any path from $a$ to $b$ and $\beta$ be any path from $b$ to $c$ . Then there is a path $\alpha\cdot\beta$ is a path from $a$ to $c$ such that $\ell(\alpha\cdot\beta)\leq\ell(\alpha)+\ell(\beta)$ . Therefore $\rho(a,c)\leq\rho(a,b)+\rho(b,c)$ finishing the proof that $\rho$ is a metric.

The metric topology induced by $\rho$ is finer than the topology of $lpc(X)$ : Suppose $U$ is an open set in $X$ (with the topology induced by $d$ ) and $C$ is some path component of $U$ . Let $x\in C$ . Find an $\epsilon$ -ball such that $B_{d}(x,\epsilon)\subseteq U$ . We claim that $B_{\rho}(x,\epsilon)\subseteq C$ : if $y\in B_{\rho}(x,\epsilon)$ , then $\rho(x,y)<\epsilon$ so there is a path $\alpha:[0,1]\to X$ from $x$ to $y$ such that $\ell(\alpha)<\epsilon$ . Since $d(x,\alpha(t))\leq\ell_t(\alpha)\leq\ell(\alpha)<\epsilon$ for all $t\in[0,1]$ , we conclude that $\alpha(t)\in B_{d}(x,\epsilon)\subseteq U$ for all $t$ . Since $\alpha$ has image in $U$ , we must have $\alpha(1)=y\in C$ , proving the claim.

The topology of $lpc(X)$ is finer than the metric topology induced by $\rho$ : For the other direction, suppose $B_{\rho}(x,\epsilon)$ is an $\epsilon$ -ball with respect to $\rho$ . Pick a point $y\in B_{\rho}(x,\epsilon)$ and let $\delta=\epsilon-\rho(x,y)$ . We claim that the path-component of $y$ in $B_{d}(y,\delta/4)$ is contained in $B_{\rho}(x,\epsilon)$ . Let $\alpha$ be a path in $B_{d}(y,\delta/4)$ such that $\alpha(0)=y$ . It suffices to check that $z=\alpha(1)\in B_{\rho}(x,\epsilon)$ . Notice that $\ell_{t}(\alpha)=d(y,\alpha(t))+d(\alpha(t),z)< \frac{\delta}{4}+\frac{\delta}{2}= \frac{3\delta}{4}$ for all $t\in[0,1].$ Thus $\ell(\alpha)\leq \frac{3\delta}{4}$ showing that $\rho(y,z)<\delta$ . We now have

$\rho(x,z)\leq\rho(x,y)+\rho(y,z)<(\epsilon-\delta)+\delta=\epsilon$

proving the claim. $\square$

Example: One thing to be wary of is that $lpc(X)$ can fail to be separable even if $X$ is a compact metric space. For instance, let $A$ be a Cantor set in $[1,2]$ . Then we can use the construction of generalized wedges of circles in the previous post to construct the planar set $X=C_A$ which is a compact metric space (and certainly separable). This is basically a wedge of circles where the circles are parameterized by a Cantor set. But $lpc(X)$ is an uncountable wedge of circles (with a metric topology – not the CW topology – at the joining point) and this is not separable. The general problem here seems to be that there might be open sets of $X$ which have uncountably many path-components!

For any given space $Y$ , $\pi_0(Y)$ will denote the set of path-components of $Y$ .

Theorem: Let $X$ be a metric space. Then $lpc(X)$ is separable if and only if $X$ is separable and $\pi_0(U)$ is countable for every open set $U\subseteq X$ .

Proof. If $lpc(X)$ is separable, then since the identity function $lpc(X)\to X$ is continuous and surjective, $X$ is separable as the continuous image of a separable space. Now pick a countable dense set $A\subset lpc(X)$ and let $U$ be a non-empty open set in $X$ . Now $\pi_0(U)$ is the set of path-components of $U$ . If $C\in\pi_0(U)$ , then $C$ is open in $lpc(X)$ and thus there is a point $a\in A\cap C$ . This gives a surjection from a subset of $A$ onto $\pi_0(U)$ showing that $\pi_0(U)$ is countable.

For the converse, if $X$ is a separable metric space then it has a countable basis $\mathscr{B}$ . Furthermore, we assume $\pi_0(B)$ is countable for every set $B\in\mathscr{B}$ . Let $\mathscr{C}=\bigcup_{B\in\mathscr{B}}\pi_0(B)$ be the collection of all path-components of the basic open sets. Then $\mathscr{C}$ is countable. If $C$ is the path-component of $x$ in an open set $U$ of $X$ , then there is a $B\in\mathscr{B}$ such that $x\in B\subseteq U.$ Now if $D$ is the path-component of $x$ in $B$ , then $x\in D\subseteq C$ where $D\in\mathscr{C}$ . This shows $\mathscr{C}$ forms a countable basis for the topology of $lpc(X)$ . Since $lpc(X)$ is metrizable (by our above work), it is also separable. $\square$

For one more post on this see: Locally path-connected coreflection (Part III)

Posted in Category Theory, General topology | Tagged coreflection, locally path-connected, metrizable | 5 Comments

The locally path-connected coreflection (Part I)

Posted on October 11, 2014 by Jeremy Brazas

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)

Posted in Algebraic Topology, Category Theory, General topology | Tagged continuous function, coreflection, coreflective subcategory, functor, locally path-connected, path-component, semi-locally simply connected, universal covering | 15 Comments

Wild Topology

The locally path-connected coreflection (Part III)

The locally path-connected coreflection (Part II)

The locally path-connected coreflection (Part I)

Some preliminary facts

A categorical interpretation

Some Examples

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The locally path-connected coreflection (Part III)

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The locally path-connected coreflection (Part II)

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The locally path-connected coreflection (Part I)

Some preliminary facts

A categorical interpretation

Some Examples

More algebraic topology

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