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 I first learned about the following general construction from some fantastic unpublished notes  of Jerzy Dydak.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 .
 Dydak, J. Coverings and fundamental groups: a new approach. Preprint. arXiv:1108.3253