## Homotopically Hausdorff Spaces II

In my first post on homotopically Hausdorff spaces, I wrote about the property which describes the existence of loops that can be deformed into arbitrarily small neighborhoods but which are not actually null-homotopic, i.e. can’t be deformed all the way back to that point. In this post, we’ll offer up different viewpoints on this property based on an approach taken from a recent paper:

Jeremy Brazas, Hanspeter Fischer, Test map characterizations of local properties of fundamental groups. Preprint. 2017. Click here for the arXiv paper.

In particular, we’ll discuss the following characterization of the homotopically Hausdorff property.

Theorem 1: For a first countable space $X$, the following are equivalent:

1. $X$ is homotopically Hausdorff,
2. Every map every map $f:\mathbb{HA}\to X$ from the harmonic archipelago induces the trivial homomorphism $f_{\#}:\pi_1(\mathbb{HA},b_0)\to\pi_1(X,f(b_0))$ on $\pi_1$.
3. For every map $g:\mathbb{H}\to X$ such that $g_{\#}([\ell_n])=1\in \pi_1(X,f(b_0))$ for every $n\geq 1$, then $f_{\#}([\ell_1\cdot\ell_2\cdots])=1$.

The harmonic archipelago $\mathbb{HA}$. Notice the Hawaiian earring $\mathbb{H}$ is a subspace and every loop based at the one wild point $b_0$ may be deformed over finitely many hills to lie within an arbitrary neighborhood of $b_0$.

Condition 2. in the theorem clarifies the notion that the harmonic archipelago really is the prototypical non-homotopically Hausdorff space since maps from it detect the same failure.

Condition 3. suggests that the homotopically Hausdorff property should be thought of as a closure property of the trivial subgroup.

Just as a reminder, here is the definition of the property we’re focusing on.

Definition: Given a path-connected space $X$ and basepoint $x_0$, we considered the subgroup

$\pi(\alpha,U)=\{[\alpha\cdot\gamma\cdot\alpha^{-}]|\gamma([0,1])\subseteq U\}\leq\pi_1(X,x_0)$

where $\alpha$ is a path starting at $x_0$ and $U$ is an open neighborhood of $\alpha(1)$. We say $X$ is homotopically Hausdorff if for every path $\alpha:[0,1]\to X$ with $\alpha(0)=x_0$, we have $\bigcap_{U\in\mathcal{T}_{\alpha(1)}}\pi(\alpha,U)=1$ where $\mathcal{T}_x$ denotes the set of all open sets in $X$ containing $x$.

### Infinite Concatenations of Paths and Homotopies

We’re going to use the Hawaiian earring as a kind of “test space” so let’s recall its construction. If $C_n$ is the circle of radius $\frac{1}{n}$ centered at $\left(\frac{1}{n},0\right)$, then $\mathbb{H}=\bigcup_{n\geq 1}C_n$ is the usual Hawaiian earring space with basepoint $b_0=(1,0)$. Let $\ell_n:[0,1]\to\mathbb{H}$ be the loop which traverses $C_n$ once in the counterclockwise direction. The homotopy classes $[\ell_n]$, $n\geq 1$ freely generate the subgroup $F=\langle [\ell_n]|n\geq 1\rangle$.

Definition: A sequence $\{\alpha_n\}_{n\geq 1}$ of paths $\alpha_n:[0,1]\to X$ is null at a point $x\in X$ if for every open neighborhood $U$ of $x$, there is an $N$ such that $\alpha_n([0,1])\subseteq U$ for all $n\geq 1$, equivalently if $\{\alpha_n\}_{n\geq 1}$ converges to the constant path at $x$.

Given a sequence of paths $\{\alpha_n\}_{n\geq 1}$ satisfying $\alpha_n(1)=\alpha_{n+1}(0)$ and which is null at $x\in X$, we may define the infinite concatenation to be the path $\alpha=\prod_{n=1}^{\infty}\alpha_n$ to be the path defined to be $\alpha_n$ on the interval $\left[1-\frac{1}{n},1-\frac{1}{n+1}\right]$ and $\alpha(1)=x$.

Sometimes, we may expand the notation as

$\displaystyle\prod_{n=1}^{\infty}\alpha_n=\alpha_1\cdot\alpha_2\cdot\alpha_3\cdots$

Infinite Concatenation

Example: An important infinite concatenation for this post will be the loop

$\ell_{\infty}=\prod_{n=1}^{\infty}\ell_n=\ell_{1}\cdot\ell_{2}\cdot\ell_{3}\cdots$

that winds once around each hoop $C_n$ of $\mathbb{H}$.

Warning: Notice here that we’re only considering infinite concatenations or “products” of loops – not homotopy classes of loops. Indeed, this operation is well defined for paths but the notion of “infinite product” of homotopy classes is not well defined in all fundamental groups.

Remark 2: What we are allowed to do with these infinite products is reparameterize them. This allows us to treat them like infinite sums and products in Calculus:

$\prod_{n=1}^{\infty}\alpha_n\simeq \left(\prod_{n=1}^{m}\alpha_n\right)\cdot\left(\prod_{n=m+1}^{\infty}\alpha_n\right)$

and so

$\left[\prod_{n=1}^{\infty}\alpha_n\right]= \left[\prod_{n=1}^{m}\alpha_n\right]\left[\prod_{n=m+1}^{\infty}\alpha_n\right]$

in the fundamental groupoid for any $m$

Proposition 3: A sequence of loops $\{\alpha\}_{n\geq 1}$ based at $x$ is null at $x$ if and only if there is a map $f:(\mathbb{H},b_0)\to (X,x)$ such that $f\circ\ell_n=\alpha_n$.

Proof. The key here is to observe that a function $f:(\mathbb{H},b_0)\to (X,x)$ is continuous if and only if $f|_{C_n}$ is continuous for each $n$ and if for every neighborhood $U$ of $x$, $f$ maps all but finitely many of the circles $C_n$ into $U$. The latter condition is clearly equivalent to the sequence of loops $f\circ\ell_n$ being null at $x$. $\square$

Lemma 4: Let $\{\alpha\}_{n\geq 1}$ be a null sequence of paths in $X$ such that $\alpha_n(0)=x$ for all $n$. Then the infinite concatenation

$\prod_{n=1}^{\infty}(\alpha_{n}\cdot\alpha_{n}^{-})=(\alpha_1\cdot\alpha_{1}^{-})\cdot(\alpha_2\cdot\alpha_{2}^{-})\cdot(\alpha_3\cdot\alpha_{3}^{-})\cdots$

is a null-homotopic loop.

Proof. Recall that for any path $\alpha$ with $\alpha(0)=x$, we can contract $\alpha\cdot\alpha^{-}$ to the constant path at $x$ by a homotopy contracting the loop back along its own image.

At height $t$, the homotopy $h(s,t)$ pictured is first $\alpha|_{[0,t]}$, constant in the black region, and then the reverse of $\alpha|_{[0,t]}$.

We construct a null-homotopy $H:[0,1]\times[0,1]\to X$ of $\prod_{n=1}^{\infty}(\alpha_{n}\cdot\alpha_{n}^{-})$ by creating an infinite concatenation of the individual contractions $h_n$ of $\alpha_n\cdot\alpha_{n}^{-}$. It will looks something like this:

Infinite concatenation of null-homotopies.

where $H$ is defined as $h_n$ on $\left[1-\frac{1}{n},1-\frac{1}{n+1}\right]\times [0,1]$. You can use the pasting lemma to verify continuity at every point except those on the right vertical wall. To verify continuity of $H$ on the right edge recall that $\{\alpha\}_{n\geq 1}$ is null at $x$. This means that given any open neighborhood $U$ of $x$, there is an $N$ such that $\alpha_n$ has image in $U$ for all $n\geq N$. But $h_n$ has image in $\alpha_n([0,1])$ for each $n$. Therefore, $H\left(\left[1-\frac{1}{N},1\right]\times [0,1]\right)\subseteq U$. We conclude that there is an open set $V$ containing $\{1\}\times [0,1]$ which is mapped into $U$ by $h$.

This verifies the continuity of $H$. $\square$

### Functorality of the Obstruction

Lemma 5: Let $f:X\to Y$ be a map, $\alpha:[0,1]\to X$ be a path, and $U$ be an open neighborhood of $f(\alpha(1))$. Then $f_{\#}(\pi(\alpha,f^{-1}(U))\leq\pi(f\circ \alpha,U)$.

Proof. If $\gamma$ is a loop in $f^{-1}(U)$, then $[\alpha\cdot\gamma\cdot\alpha^{-}]$ is a generic element of $\pi(\alpha,f^{-1}(U))$. Since $f\circ \gamma$ has image in $U$, it follows that

$f_{\#}([\alpha\cdot\gamma\cdot\alpha^{-}])=[(f\circ\alpha)\cdot(f\circ\gamma)\cdot(f\circ \alpha)^{-}]\in\pi(f\circ \alpha,U)$. $\square$

Corollary 6: Let $f:X\to Y$ be a map, $\alpha:[0,1]\to X$ be a path from $x_0$ to $x$, and set $f(x_0)=y_0$ and $f(x)=y$. Then

$f_{\#}\left(\bigcap_{V\in\mathcal{T}_x}\pi(\alpha,V)\right)\leq\bigcap_{U\in\mathcal{T}_y}\pi(f\circ\alpha,U)$

as subgroups of $\pi_1(Y,y_0)$.

Proof. Suppose $g\in\pi(\alpha,V)$ for all $V\in\mathcal{T}_x$ and pick any $U\in\mathcal{T}_y$. Then $g\in\pi(\alpha,f^{-1}(U))$ and by Lemma 5, we have $f_{\#}(g)\in\pi(f\circ\alpha,U)$. Thus $f_{\#}(g)\in\bigcap_{U\in\mathcal{T}_y}\pi(f\circ\alpha,U)$. $\square$

Interpretation: Corollary 6 can be thought of as saying that the “obstruction” subgroups $\bigcap_{U\in\mathcal{T}_x}\pi(\alpha,U)$ which detect the failure of the homotopically Hausdorff property are functorial since continuous maps induce homomorphisms that always map obstruction subgroups into obstruction subgroups.

### (1. $\Rightarrow$ 2.)

In the harmonic archipelago $\mathbb{HA}$ every loop based at $b_0$ may be continuously deformed within an arbitrary neighborhood $U$ of the basepoint $b_0$. Thus if $\alpha:[0,1]\to\mathbb{HA}$ is the constant path at the wild point $b_0$, then $\pi(\alpha,U)=\pi_1(\mathbb{HA},b_0)$ for every open neighborhood $U$ of $b_0$. Hence $\bigcap_{U\in\mathcal{T}_{b_0}}\pi(\alpha,U)=\pi_1(\mathbb{HA},b_0)$.

Now suppose $f:\mathbb{HA}\to X$ is a map such that the induced homomorphism $f_{\#}:\pi_1(\mathbb{HA},b_0)\to\pi_1(X,f(b_0))$ is not the trivial homomorphism, then by Corollary 6, we have

$1\neq f_{\#}(\pi_1(\mathbb{HA},b_0))=f_{\#}\left(\bigcap_{V\in\mathcal{T}_{b_0}}\pi(\alpha,V)\right)\leq\bigcap_{U\in\mathcal{T}_{f(b_0)}}\pi(f\circ\alpha,U)$.

Now $f\circ \alpha$ is a constant path at $f(b_0)$ such that $\bigcap_{U\in\mathcal{T}_{f(b_0)}}\pi(f\circ\alpha,U)\neq 1$, which means $X$ cannot be homotopically Hausdorff.

Note: this direction of Theorem 1 doesn’t actually require first countability.

### (2. $\Rightarrow$ 3.)

The main fact that we need is that if $j:\mathbb{H}\to\mathbb{HA}$ is the inclusion map, then $j_{\#}([\ell_{\infty}])\neq 1$ in $\pi_1(\mathbb{HA},b_0)$. I give a simple explicit proof of this fact in this post. A quick reminder of how this is done: compactness of the unit disk means that a null-homotopy of $j\circ\ell_{\infty}$ can only intersect finitely many of the hills of $\mathbb{HA}$. So if $E_n$ is the interior of the n-th hill, then $j\circ\ell_{\infty}$ is null-homotopic in $\mathbb{H}\cup\bigcup_{1\leq n\leq N}E_n$ for some $N$ but this is impossible since $\ell_{\infty}$ winds around the circle retracts $C_n$, $n>N$ in a non-trivial way.

We prove the contrapositive. Suppose 3. does not hold. Then there exists a map $g:\mathbb{H}\to X$ such that $g_{\#}([\ell_n])=1$ for all $n\geq 1$ and $g_{\#}([\ell_{\infty}])\neq 1$.

For each $n\geq 1$, the loop $g\circ(\ell_n\cdot\ell_{n+1}^{-}):S^1\to X$ is null-homotopic loop in $X$ and therefore extends to a map on the unit disk. Since each of the holes in $\mathbb{H}$ can be extended to “large” disks, $g$ extends to a map $f:\mathbb{HA}\to X$ such that $f\circ j=g$. So we have $[j\circ\ell_{\infty}]\neq 1$ and $f_{\#}([j\circ\ell_{\infty}])=g_{\#}([\ell_{\infty}])\neq 1$. Therefore $f_{\#}:\pi_1(\mathbb{HA},b_0)\to\pi_1(X,f(b_0))$ is not the trivial homomorphism.

Note: This part of the proof does not require first countability either.

### (3. $\Rightarrow$ 2.)

For this direction of Theorem 1, we do need the assumption that $X$ is homotopically Hausdorff. We prove the contrapositive.

Suppose that $X$ is first countable and that $X$ fails to be homotopically Hausdorff.  Then there exists a path $\alpha:[0,1]\to X$ from $x_0$ to $x$ and a loop $\gamma$ based at $x$ such that

$1\neq[\alpha\cdot\gamma\cdot\alpha^{-}]\in\bigcap_{U\in\mathcal{T}_x}\pi(\alpha,U)\leq\pi_1(X,x_0)$.

Notice that if we conjugate by $[\alpha]^{-1}$, then we see that $1\neq [\gamma]\in\pi_1(X,x)$.

Let $U_1\supset U_2\supset U_3\supset...$ be a countable neighborhood base at $x$. Then

$1\neq[\alpha\cdot\gamma\cdot\alpha^{-}]\in\bigcap_{n\geq 1}\pi(\alpha,U_n)\leq\pi_1(X,x_0)$.

Hence, for each $n\geq 1$, there is a loop $\gamma_n:[0,1]\to U_n$ based at $x$ such that $[\alpha\cdot\gamma\cdot\alpha^{-}]=[\alpha\cdot\gamma_n\cdot\alpha^{-}]$. In particular, $[\gamma]=[\gamma_n]$  in $\pi_1(X,x)$ for each $n\geq 1$.

By construction, the sequence of loops $\{\gamma_n\}_{n\geq 1}$ is null at $x$. Therefore, the sequence $\{\gamma_n\cdot\gamma_{n+1}^{-}\}_{n\geq 1}$ of loops is also null at $x$. Using Proposition 3, we put this sequence together to construct a continuous function $g:(\mathbb{H},b_0)\to (X,x)$ defined by $g\circ\ell_n=\gamma_n\cdot\gamma_{n+1}^{-}$.

Notice that $g_{\#}([\ell_n])=[\gamma_n][\gamma_{n+1}]^{-1}=[\gamma][\gamma]^{-1}=1$ for each $n\geq 1$.

Now, we use a “telescoping product” to prove that $g_{\#}([\ell_{\infty}])\neq 1$.

We have

$g_{\#}([\ell_{\infty}])=[g\circ\ell_{\infty}]=\left[\prod_{n=1}^{\infty}(\gamma_{n}\cdot\gamma_{n+1}^{-})\right]=[\gamma_1]\left[\prod_{n=2}^{\infty}(\gamma_n\cdot\gamma_{n}^{-})\right]$

where the last equality is allowed according to Remark 2. But $\left[\prod_{n=2}^{\infty}(\gamma_n\cdot\gamma_{n}^{-})\right]=1$ by Lemma 4.

Therefore $g_{\#}([\ell_{\infty}])=[\gamma](1)=[\gamma]\neq 1$. This completes the proof! $\square$

### Takeaway

There are a few things I hope you can take away from this post. Ultimately, we have taken this important obstruction and teased it apart into different viewpoints. To me, that makes good mathematics.

1. Because the abelianization of $\pi_1(\mathbb{HA},b_0)$ is isomorphic to $\prod_{n=1}^{\infty}\mathbb{Z}/\bigoplus_{n=1}^{\infty}\mathbb{Z}$ (a highly non-trivial fact), Condition 2. in Theorem 1 looks like a non-abelian generalization of the cotorsion free property defined for abelian groups. In fact, a direct Corollary of 1. $\Leftrightarrow$ 2. is that if $\pi_1(X,x_0)$ is abelian and cotorsion free, then $X$ is homotopically Hausdorff. Not necessarily groundbreaking, but the connection is certainly noteworthy.
2. Condition 3 in Theorem 1 looks like a closure property – something like subgroup-closure under infinite products…we make this more precise and apply the idea widely in the paper I shamelessly plug at the top of the post.

## Homotopically Hausdorff Spaces I

In previous posts, I wrote about the harmonic archipelago $\mathbb{HA}$ (see also here and here):

Harmonic Archipelago

as well as the Griffiths Twin Cone $\mathbb{G}$.

Griffiths Twin Cone

One special feature of these 2-dimensional spaces is that any loop either of these spaces can be deformed to lie within an arbitrarily small neighborhood of the basepoint. In fact, these are the prototypical spaces for this pathology. The existence of non-trivial loops that can be deformed into arbitrarily small neighborhoods can be thought of as an obstruction to applying covering space and shape theoretic techniques to understand the fundamental group. It turns out there is a named property that gets to the heart of this obstruction.

It’s actually an open question whether or not $\mathbb{HA}$ and $\mathbb{G}$ have isomorphic fundamental groups! They are known to have isomorphic first singular homology groups. The difficulty of this question stems from the fact that they are not homotopically Hausdorff.

### History

This property appeared in two sets of unpublished notes before it appeared in a published paper with the now-standard name.

1. W.A. Bogley, A.J. Sieradski, Universal path spaces, Unpublished notes. 1998
• homotopically Hausdorff is equivalent to the author’s notation of the trivial subgroup being “totally closed.”
2. A. Zastrow, Generalized $\pi_1$-determined covering spaces, Unpublished notes. 2002.
• homotopically Hausdorff is equivalent to what the author calls “weak $\pi_1$-continuity”
3. J.W. Cannon, G.R. Conner, On the fundamental groups of one-dimensional spaces, Topology Appl. 153 (2006) pp. 2648-2672.
• the introduction of the term “homotopically Hausdorff”

Since it’s introduction, this fundamental property has appeared in a large number of publications.

### Some Important Subgroups

Assumptions: $X$ will be a path-connected Hausdorff space and $x_0\in X$ will be a basepoint.

Definition: Given a path $\alpha:[0,1]\to X$ with $\alpha(0)=x_0$ and an open neighborhood $U$ of $\alpha(1)$, let

$\displaystyle \pi(\alpha,U)=\{[\alpha\cdot\gamma\cdot\alpha^{-}]\in \pi_1(X,x_0)|\gamma([0,1])\subseteq U\}$.

We can describe $\pi(\alpha,U)$ as the subgroup of $\pi_1(X,x_0)$ consisting of “lolipop” loops that go out on the fixed path $\alpha$, move around in $U$, and then go back along the reverse of $\alpha$.

An element of $\pi(\alpha,U)$

Definition: Given $x\in X$ and a neighborhood $U$ of $x$, let

$\displaystyle \pi(x,U)=\langle \pi(\alpha,U)|\alpha(1)\in U\rangle\leq \pi_1(X,x_0)$

to be the subgroup generated by all $\pi(\alpha,U)$ where $\alpha$ ranges over all paths from $x_0$ to $x$. This means a generic element of $\pi(x,U)$ is of the form $\displaystyle\prod_{i=1}^{n}[\alpha_i\cdot\gamma_i\cdot\alpha_{i}^{-}]$ where all the loops $\gamma_{i}$ have image in $U$.

An element of $\pi(x,U)$

Observation: For any loop $\beta$ based at $x_0$, $[\beta]\pi(\alpha,U)[\beta]^{-1}]=\pi(\beta\cdot\alpha,U)$.

Observation: $\pi(\alpha,U)\leq\pi(x,U)$ whenever $\alpha(1)=x$.

Notational Remark: The notation for $\pi(\alpha,U)$ and $\pi(x,U)$ is influenced by E.H. Spanier’s excellent Algebraic Topology textbook.

Proposition: For any $x\in X$, the subgroup $\pi(x,U)\trianglelefteq\pi_1(X,x_0)$ is a normal subgroup of $\pi_1(X,x_0)$.

Proof. If $[\beta]\in\pi_1(X,x_0)$ and $\pi(x,U)$ is of the form $\displaystyle \prod_{i=1}^{n}[\alpha_i\cdot\gamma_i\cdot\alpha_{i}^{-}]$ is a generic element of $\pi(x,U)$ where each loop $\gamma_i$ has image in $U$, then

$[\beta]\left(\prod_{i=1}^{n}[\alpha_i\cdot\gamma_i\cdot\alpha_{i}^{-}]\right)[\beta]^{-1}=\prod_{i=1}^{n}[\beta\cdot\alpha_i\cdot\gamma_i\cdot\alpha_{i}^{-}\cdot\beta^{-}]$,

which is an element of $\pi(x,U)$. $\square$

### Defining the homotopically Hausdorff property

Definition: If $x\in X$, let $\mathcal{T}_x$ be the set of open neighborhoods in $X$ containing $x$. We say a space $X$ is homotopically Hausdorff at $x\in X$ if for every path $\alpha$ from $x_0$ to $x$, we have

$\displaystyle \bigcap_{U\in\mathcal{T}_x}\pi(\alpha,U)=1$

where $1$ denotes the trivial subgroup. We say $X$ is homotopically Hausdorff if $X$ is homotopically Hausdorff at all of its points.

Intuition: Recall that $\gamma$ is a null-homotopic loop based at $x$ if and only if $\alpha\cdot\gamma\cdot\alpha^{-}$ is a null-homotopic loop based at $\alpha(0)$. So a space $X$ fails to be homotopically Hausdorff if there is a point $x\in X$ and a non-null homotopic loop $\gamma$ based at $x$ which may be homotoped within an arbitrary neighborhood of $x$.

So what happens if $X$ is not homotopically Hausdorff? It means that there is a point $x\in X$, a path $\alpha$ from $x_0$ to $x$, and a non-null-homotopic loop $\gamma$ based at $x$ such that the class $[\alpha\cdot\gamma\cdot\alpha^{-}]$ can be represented by $\alpha\cdot\gamma\cdot\alpha^{-}$ where $\gamma$ may be chosen to have image in an arbitrary neighborhood of $x$.

The conjugating loops $\alpha$ are simply the way of describing this property as ranging over all points $x\in X$ while still using a fixed basepoint $x_0$. We could have defined it without them, but there is also a subgroup-relative version of the homotopically Hausdorff property for which these conjugating paths are necessary.

Indeed, the harmonic archipelago and Griffiths twin cone spaces are not homotopically Hausdorff. It turns out that many spaces are homotopically Hausdorff though. Obvious ones include spaces that admit a simply connected covering space (including manifolds, CW-complexes, etc.). Note the following doesn’t actually require local path connectivity.

Definition: We say $X$ is semilocally simply connected at $x\in X$ if there exists an open neighborhood latex $V$ of $x$ such that the inclusion $i:V\to X$ induces the trivial homomorphism $i_{\#}:\pi_1(V,x)\to\pi_1(X,x)$, i.e. if every loop in $V$ based at $x$ is null-homotopic in $X$ by a (possibly large) homotopy in $X$. We say $X$ is semilocally simply connected if it is semilocally simply connected at all of its points.

Observation: A space $X$ is semilocally simply connected at $x$ if and only if there is an open neighborhood $V$ of $x$ such that $\pi(x,V)=1$. In this case, for every $\alpha$ with $\alpha(1)=x$, we have

$\displaystyle\bigcap_{U\in\mathcal{T}_x}\pi(\alpha,U)\leq \bigcap_{U\in\mathcal{T}_x}\pi(x,U)\leq\pi(x,V)=1$

Hence, semilocally simply connected $\Rightarrow$ homotopically Hausdorff.

In fact, there are many important intermediate properties to explore…perhaps later.

Corollary: If $X$ admits a simply connected covering space, then $X$ is homotopically Hausdorff.

Proof. It’s a nice exercise to show that every such space $X$ is semilocally simply connected. This does not require any local path connectivity assumptions.

The homotopically Hausdorff property is actually much weaker than the semilocally simply connected property but it is a necessary property to have in order to applying generalized covering space theories. Many, many other spaces without traditional universal covers are also homotopically Hausdorff, including all 1-dimensional spaces (e.g. Hawaiian earring, Menger Sponge, etc.), all planar spaces, and many (but not all) 2-dimensional spaces, among others.

### Why is “Hausdorff” in the name?

The name suggests that there is some kind of separation-like axiom here. Indeed, if a space is homotopically Hausdorff, then we can separate homotopy classes in a certain topological sense.

The standard universal covering space construction: Let $\widetilde{X}$ be the set of homotopy (rel. endpoint) classes of paths in $X$ starting at $x_0$. We give this set the standard topology which is sometimes called “whisker topology”. A basic open set generating the standard topology is of the form

$B([\alpha],U)=\{[\alpha\cdot\gamma]|\gamma([0,1])\subseteq U\}$

where $U$ is an open neighborhood of $\alpha(1)$.

An element in $B([\alpha],U)$. Such an element can only differ from $[\alpha]$ at its terminal end but there it may be a complicated extension within $U$.

It’s a nice exercise in covering space theory to show that these sets form a basis for a topology on $\widetilde{X}$. Recent work of my own actually shows that this topology is the only topology of generalized covering space theories for locally path-connected spaces – any other notion of generalized covering space based on homotopy-lifting must be equivalent to it. Here is the reasoning for the name.

Theorem: The following are equivalent:

1. $X$ is homotopically Hausdorff,
2. $\widetilde{X}$ is Hausdorff ($T_2$),
3. $\widetilde{X}$ is $T_1$,
4. $\widetilde{X}$ is $T_0$.

Proof. 1. $\Rightarrow$ 2. Suppose $\widetilde{X}$ is not Hausdorff. Then there are paths $\alpha,\beta:[0,1]\to X$ starting at $x_0$ such that $[\alpha]$ and $[\beta]$ are distinct homotopy classes which cannot be separated by disjoint basic open sets in $\widetilde{X}$. Since $X$ is assumed to be Hausdorff, if $\alpha(1)\neq \beta(1)$, then we could separate $[\alpha]$ and $[\beta]$ in $\widetilde{X}$ simply by taking $B([\alpha],U)$, $B([\beta],V)$ where $U\cap V=\emptyset$. Thus we must have $x=\alpha(1)=\beta(1)$. Notice $[\alpha\cdot\beta^{-}]\neq 1$ in $\pi_1(X,x_0)$. Let $U$ be an arbitrary open neighborhood of $x$. By assumption, $B([\alpha],U)\cap B([\beta],U)\neq \emptyset$ so we have $[\alpha\cdot\delta]=[\beta\cdot\epsilon]$ for paths $\delta,\epsilon$ in $U$. Since $[\beta]=[\alpha\cdot\delta\cdot\epsilon^{-}]$, we have

$[\alpha\cdot\beta^{-}]=[\beta\cdot\alpha^{-}]^{-1}= ([\alpha\cdot\delta\cdot\epsilon^{-}\cdot\alpha^{-}])^{-1}=[\alpha\cdot\epsilon\cdot\delta^{-}\cdot\alpha^{-}]$.

This means $[\alpha\cdot\beta^{-}]\in \pi(\alpha,U)$. Since $U$ was arbitrary, it follows that $1\neq [\alpha\cdot\beta^{-}]\in \bigcap_{U\in\mathcal{T}_x}\pi(\alpha,U)$. Thus $X$ is not homotopically Hausdorff.

2. $\Rightarrow$ 3. $\Rightarrow$ 4. are basic facts of separation axioms.

4. $\Rightarrow$ 1. Suppose $X$ is not homotopically Hausdorff. Then there is a path $\alpha$ from $x_0$ to $\alpha(1)=x$ and an element $1\neq [\beta]\in\bigcap_{U\in\mathcal{T}_x}\pi(\alpha,U)$. Since $[\beta]\neq 1$, we have $[\alpha]\neq[\beta\cdot\alpha]$. Let $B([\alpha],U)$ be a basic neighborhood of $[\alpha]$ in $\widetilde{X}$. By choice of $[\beta]$, we may write $[\beta]=[\alpha\cdot\gamma\cdot\alpha^{-}]$ where $\gamma$ is a loop in $U$. Thus $[\beta\cdot\alpha]=[\alpha\cdot\gamma]\in B([\alpha],U)$. Since $[\beta\cdot\alpha]$ lies in every neighborhood of $[\alpha]$, the two are topologically indistinguishable in $\widetilde{X}$, i.e. $\widetilde{X}$ is not $T_0$. $\square$

## The locally path-connected coreflection III

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 $\coprod_{j\in J}X_j$ 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 $f(x)\in C$ to $v$. Since $C$ is the path component of $f(x)$ in $U$, $q\circ\alpha$ 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

1. $U\cap X_j$ is open in $X_j$ for every $j\in J$.
2. 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 Hawaiian earring.

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