This post gives another look at the locally path-connected coreflection that I’ve found quite interesting and useful. We’re motivated by the following two basic observations.

**Lemma 1:** The topological sum of a family of locally path-connected spaces is locally path-connected.

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

*Proof.* Let be a quotient map where is locally path-connected. Let be an open set in and be a non-empty path-component of . It suffices to check that is open in . Since is quotient, we need to check that is open in . If , then there is an open, path connected neighborhood such that . We claim that . Let and be a path from to . Then is a path from to . Since is the path component of in , must have image entirely in . Thus has image in , in particular .

denotes the usual category of topological spaces and continuous functions. Here is an important definition from Categorical Topology.

**Definition 3:** Let be a subcategory of . The *coreflective hull* of is the full subcategory of consisting of all spaces which are the quotient of a topological sum (i.e. disjoint union) of spaces in .

Certainly . Moreover, the coreflective hull is the category “generated” by the collection of spaces in the following sense: the inclusion has a right adjoint (called a *coreflection*) . In particular, is the space with the same underlying set as but a set is open if and only if is open in for every map where . In short, has the final topology with respect to the collection of all maps from spaces in to .

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

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 I first learned about the following general construction from some fantastic unpublished notes [1] of Jerzy Dydak.

**Definition 4:** Let be a directed set. The *directed wedge* of a collection of spaces indexed by is the wedge sum (given by identifying the basepoints to a single point ) with the following topology: A set is open if and only if

- is open in for every .
- if , then there is a such that for all .

In particular if is the unit interval for every , then is the *-arc-hedgehog*.

**Example 5:** If is the natural numbers, then the -arc hedgehog space is the space of a sequence of shrinking intervals joined at a point.

In the case that and is the unit circle. The directed wedge is the Hawaiian earring.

**Lemma 6:** If is a collection of path-connected and locally path-connected spaces, then is path-connected and locally path-connected.

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

**Theorem 7:** Let be the subcategory of all -arc hedgehogs. Then .

*Proof.* Since every arc-hedgehog is locally path-connected, have and thus . For the other inclusion, suppose is locally path-connected. Suppose . Clearly if is open, then is open in for every map . For the converse, suppose is not open. There exists a point such that for every path-connected neighborhood of , there is a point . Let be the directed set of path-connected neighborhoods of . For each , find a path from to . Define a map so that the restriction to the -th arc is the path . It is easy to see that is continuous based on how we defined the topology of . Since , we have , however, if denotes the end of the -th arc , then . Thus for all . It follows that cannot be open since is a net in converging to the joining point .

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

**References: **

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