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:

generalized wedge of circles

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.

path components of a neighborhood

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.

the lpc-coreflection of a generalized wedge of circles

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.

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3 Responses to The locally path-connected coreflection

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