Topological Group Reflections: turning a group with topology into a topological group, Part I

Posted on September 19, 2024 by Jeremy Brazas

The main complicating factor with the quotient topology on the fundamental group \pi_{1}^{qtop}(X,x_0) is that it often fails to be a topological group. More generally, if M is a topological monoid and the quotient space G=M/\mathord{\sim} is group by a relation that preserves the operation, then G need not be a topological group. From one viewpoint, this could be considered a disappointing fact. So what causes this failure to happen? It turns out that this is a fairly deep topological question that I don’t have a complete answer too. It is partially due to some issues with the general topological category.

At this point, some folks may be inclined to point out that this failure doesn’t occur if one works internal to some “nicer” category like the category of k-spaces. This is a fair point. I agree that there are many situations in which the k-space category is more suitable and that one can obtain a group-object for \pi_1 in k-spaces. However, most folks who are adamant about forcing things into a “convenient” category (typically in AT) are uncomfortable outside of such categories and are not used to dealing with objects where the actual topology matters. To insist that a convenient coreflection is the only reasonable “fix” is short-sighted and closes the door to good mathematics. It turns out that forcing things into the k-space category loses too much information that is actually important to topologized fundamental groups and what they are good at detecting. If I had only stuck with k-spaces and been closed off to considering more challenging topological options, I wouldn’t have come up with the mathematics for what has (weirdly) become my most popular paper and led to surprising applications in topological group theory.

In this post, I’m going to talk about the actual “topological fix” to the quotient topology of \pi_{1}^{qtop}(X,x_0). We’re going to decide if there exists a functorial topology on \pi_1(X,x_0) that gives a topological group while retaining as much of the original information of the quotient topology as possible. Along the way, we’ll explore some interesting mathematical structures and categorical topology.

Let’s approach this from a very general viewpoint. Say I have a group G with identity element e\in G and some topology T on G. The pair (G,T) might already be a topological group. If so, great! Leave it alone. But maybe it’s not. Maybe the group operation doesn’t interact nicely with the topology at all. Specifically, the group operation \mu:G\times G\to G, \mu(a,b)= ab and group inversion i:G\to G, i(a)= a^{-1} are NOT assumed to be continuous. But let’s see if we can modify the topology T, with minimal information loss, so that we end up with a topological group. The underlying group will be identical to the original group. It’s only the topology that’s going to change.

Let \sigma:G\times G\to G be the function \sigma(a,b)=ab^{-1}. Notice that if \sigma is continuous, then inversion i(b)=\sigma(e,b)=b^{-1} is continuous and group multiplication \mu(a,b)=\sigma(a,i(b))=ab is continuous. Thus G is a topological group. Conversely, if G is a topological group, then \sigma(a,b)=\mu(a,i(b)) is continuous.

Proposition 1. Let G be a group with topology. Then the following are equivalent:

  1. G is a topological group ,
  2. \sigma:G\times G\to G is continuous,
  3. \sigma:G\times G\to G is a quotient map,
  4. \sigma:G\times G\to G is a continuous open surjection.

Proof. We’ve already noted the equivalence of 1. and 2. Certainly 4 \Rightarrow 3 \Rightarrow 2 from basic topology. That 2 implies 4 is an introductory excercise in topological group theory. \square

Since the function \sigma tells us, in a sense, how close T is to being a topological group, we can “force” it to be continuous by changing the topology of the codomain.

Given a group G equipped with topology T_0, let c(G) be the same group as G but equipped with the quotient topology inherited from \sigma, i.e. so that \sigma:G\times G\to c(G) is a quotient map. Let T_1 denote this quotient topology. Thus U\in T_1 if and only if \sigma^{-1}(U) is open in G\times G. The identity function G\to c(G) can be defined by \sigma(g,e)=g and is is continuous. Hence, the topology T_1 of c(G) is coarser than that of G. Also, according to Proposition 1, we have c(G)=G if and only if G is a topological group.

So we fixed it right? Isn’t c(G) a topological group? Well…that’s not immediately clear. We’d have to check that \sigma:c(G)\times c(G)\to c(G) is continuous and there does not seem to be an easy way to prove this. We only know that \sigma:G\times G\to c(G) is continuous. Even so, it seems that we have at least moved in the right direction since we removed some open sets from T_0 to get the topology T_1 but not too many open sets.

So let’s do it again. Let c^2(G) be the same group as G but equipped with the quotient topology inherited from \sigma with domain c(G)\times c(G), i.e. so that \sigma:c(G)\times c(G)\to c^2(G) is a quotient map. Let T_2 denote this quotient topology. Now, c(G) is a topological group if and only if c^2(G)=c(G). If this is the case, we may stop. If not, we try it again.

Since G\mapsto c(G) is a general operation we can perform for any group with any topology, we could define c^{n+1}(G)=c(c^{n}(G)) recursively for all n\geq 1 where T_n denotes the topology of c^{n}(G). We are left with proper inclusions of topologies T_0\supseteq T_1\supseteq T_2\supseteq T_3\supseteq \cdots. This sequence will stabilize if and only if c^{n}(G) is a topological group for some n. However, there is nothing promising that this will happen. So regular induction may fail us… We cry a few tears and then remember that there is, in fact, a way to continue this process to persist past infinity. This is where we are ever so grateful for

transfinite induction.

We have so far defined the induction for finite ordinals. The first infinite ordinal is \omega=\{0,1,2,3,\dots\}. Let c^{\omega}(G) be the same group as G but with topology T_{\omega}=\bigcap_{n=0}^{\infty}T_n. Notice that T_n\supseteq T_{\omega} for all n\in\omega. The reason that this intersection makes sense (besides being a natural choice) is because the descending sequence T_0\supseteq T_1\supseteq T_2\supseteq T_3\supseteq \cdots of topologies is telling us that to end up with a topological group we must remove the open sets from T_0\backslash T_1 and then from T_1\backslash T_2 and so on. If we remove all of these open sets, we’re only left with those in \bigcap_{n\geq 0}T_n.

Most transfinite induction arguments break into two cases – the successor ordinal case (a single step) and limit ordinal case (moving past infinitely many steps). The construction above provides a template for how to do the general transfinite induction: assume c^{0}(G)=G. Suppose \lambda is an ordinal and c^{\kappa}(G) has been defined as the group G with topology T_{\kappa} for every ordinal \kappa<\lambda.

Successor ordinal case: Suppose \lambda=\kappa+1 for ordinal \kappa. Since c^{\kappa}(G) is assumed to be defined, we may let c^{\lambda+1}(G)=c(c^{\kappa}(G)), that is, the same underlying group as G but with the quotient topology inherited from \sigma with domain c^{\kappa}(G)\times c^{\kappa}(G), i.e. so that \sigma: c^{\kappa}(G)\times c^{\kappa}(G)\to c^{\lambda}(G) is a quotient map. Let T_{\lambda} denote the topology of c^{\lambda}(G).

Limit ordinal case: Suppose \lambda is a limit ordinal. Our induction hypothesis is that c^{\kappa}(C) is defined with topology T_{\kappa} for all \kappa<\lambda. Thus we let c^{\lambda}(G) be the same group as G but with the topology T_{\lambda}=\bigcap_{\kappa<\lambda}T_{\kappa}.

This transfinite induction results in a nested sequence of topologies all on the same group G:

T_0\supseteq T_1\supseteq T_2 \supseteq \cdots T_{\omega}\supseteq T_{\omega+1}\supseteq T_{\omega+2}\supseteq\cdots\supseteq T_{\omega+\omega}\supseteq\cdots\supseteq T_{\omega^{\omega}}\supseteq\cdots T_{\omega_1}\supseteq \cdots

where I interjected some other infinite ordinals for good measure. But this sequences goes on for all ordinals. 

Proposition 2: c^{\lambda}(G) is a topological group if and only if T_{\kappa}=T_{\kappa+1}.

Proof. If c^{\kappa}(G) is a topological group, then \sigma: c^{\kappa}(G)\times c^{\kappa}(G)\to c^{\kappa}(G) is continuous and is therefore a quotient map. But quotient topologies are unique and the above construction gives that \sigma: c^{\kappa}(G)\times c^{\kappa}(G)\to c^{\kappa+1}(G) is a quotient map. Hence, T_{\kappa}=T_{\kappa+1}. Conversely, suppse T_{\kappa}=T_{\kappa+1}. But \sigma: c^{\kappa}(G)\times c^{\kappa}(G)\to c^{\kappa+1}(G) is continuous by construction and since we are assuming c^{\kappa}(G)=c^{\kappa+1}(G), we make this replacement in the codomain to see that \sigma: c^{\kappa}(G)\times c^{\kappa}(G)\to c^{\kappa}(G) is continuous. Hence, c^{\kappa}(G) is a topological group. \square

This proposition tells us that the transfinite sequence \{c^{\kappa}(G)\}_{\kappa} of groups with topology can only stabilize if it stabilizes at a topological group. So must it stabilize?

Proposition 3: The transfinite sequence \{c^{\kappa}(G)\}_{\kappa} of groups with topology stabilizes at some ordinal \kappa.

Proof. Here, we have to appeal to cardinality. Suppose that the transfinite sequence of topologies \{T_{\kappa}\}_{\kappa} never stabilizes. This means for each ordinal \kappa, T_{\kappa+1} is a proper subset of T_{\kappa} and all of the sets T_{\kappa}\backslash T_{\kappa+1} are pairwise-disjoint in the original topology T_0. In other words, for each ordinal \kappa, we removed some at least one set U_{\kappa}\in T_{\kappa}\backslash T_{\kappa+1} as we proceeded through the induction.

But the ordinal numbers go really far…too far. If you give me any set like T_0, there exists an ordinal number \lambda with a cardinality greater than that of T_0. But the function f:\lambda\to T_0, f(\kappa)=U_{\kappa} is an injection and this is a contradiction because we can’t inject \lambda into T_0 since we found \lambda so that |\lambda|>|T_0|. \square

Putting everything together, we get the following.

Theorem 4: The transfinite sequence of topologies \{c^{\kappa}(G)\}_{\kappa} stabilizes at some ordinal. In particular, the transfinite sequence \{c^{\kappa}(G)\}_{\kappa} of groups with topology stabilizes at a topological group, denoted \tau(G).

The construction G\mapsto \tau(G) takes in a group with topology and outputs a topological group in “the most efficient way possible.” Formally, the topology of \tau(G) is the finest topology on the group that is (1) coarser than that of G and (2) which makes the group a topological group.

Did we really need to go through all this trouble to build \tau(G)? Well…there are other ways we could choose to show that topology of \tau(G) exists but none of them are going to have simple descriptions. Because, remember, a union of topologies need not be a topology. So we can’t build the topology from below. We really do have to carefully wittle down the topology of G without removing too much information.

My bolded claim is not exactly proved yet, but this and many other things we can say about the construction become easier to prove once we show that \tau is a functor. We’ll talk about maps in Part II.

 

 

 

 

 

Posted in Categorical Topology, quasitopological groups, quotient topology, reflection functor, Topological groups, Uncategorized | 4 Comments

Local n-connectivity vs. the LC^n Property, Part 2

Posted on October 17, 2023 by Jeremy Brazas

In Part 1, I defined local n-connectivity, which says that a space has a basis of n-connected open sets, and the LC^n porperty, which says that for all 0\leq k\leq n, “small” maps from the k-sphere contract using “small” homotopies of “relatively the same size.” The difference is subtle and often motivates the use of LC^n over local n-connectivity in metric geometry, shape theory, and wild topology.

In Part 1, I went ahead and explained the difference for paths, i.e. in dimension n=0. In that case, the pointwise or based definitions (having the property at a given point) differ but the global properties (having the local property at all points) are equivalent: Locally 0-connected \Leftrightarrow LC^0. I also pointed out some examples that showed things can get a bit wild if you don’t assume first countability. In this post, we’ll explore the higher dimensional situation, i.e. where n\geq 1.

In dimension n=1, I learned about this difference from this MathOverflow question asked back in 2019 which is really asking for an example of a LC^1 space that is not locally 1-connected. Patrick Gillespie (now a PhD student at UTK and who has been a guest author on this blog) was in my topology course and asked about it. I told the class I’d offer extra credit to anyone who solved it. We had several discussions about it and Patrick came up with the space appearing in Figure 1, which I illustrated using Mathematica:

Figure 1: An example of a LC^1 space that is not locally 1-connected. Although it is not pictured, you want to imagine the large central circle filled in with a disk in this image. It becomes more difficult to see the other levels when it is included in the picture.

Construction: To build this space explicitly, the idea is to glue together a sequence of shrinking disks together in a certain way. Actually, it’s easier to visualize if you think of these disks as cones over circles. For m\in\mathbb{N}, let A_m be a circle with basepoint a_m and \alpha_m:[0,1]\to A_m be a loop based at a_m that parameterizes the circle. Let CA_m=[0,1]\times A_m/\{1\}\times A_m be the cone over A_m. We take the image of (0,a_m) to be the basepoint of CA_m and we identify A_m with the “base” of the cone, i.e. the image of A_m\times\{0\}. Let \ell_m:[0,1]\to CA_m be the path that runs up the “spine” of the cone, that is where \ell_m(t) is the image of (a_m,t) in CA_m.

Now, consider the wedge sum/based one-point union X=\bigvee_{n\geq 1}CA_m where the points a_m are identified to a single wedge-point x_0. We give X the topology of a shrinking wedge determined by the following: U\subseteq X is open if and only if U\cap CA_m is open in CA_m for all m and if whenever x_0\in U, we have CA_m\subseteq U for all but finitely many m (see Figure 2).

Figure 2: A shrinking one-point union of cones over the the circle. The base circles form an infinite earring and the spines of each cone runs up to the vertex.

Remark 1: X is contractible by a null-homotopy that fixes the basepoint x_0. To construct a contraction of X one only need glue together null-homotopies of each cone CA_m that leaves the basepoint of CA_m (on the base circle) fixed.

Now we do some gluing. Recall that \alpha_m:[0,1]\to A_m runs around the m-th base circle and \ell_m:[0,1]\to CA_m runs up the m-th spine (these are both bolded in Figure 2). Let \sim be the smallest relation on X generated by \ell_m(t)\sim\alpha_{m+1}(t) for all m\geq 1 and t\in[0,1]. The resulting quotient space Y=X/\mathord{\sim} the 2-dimensional Peano continuum Y from Figure 1 but where the large central circle is filled in by a disk or formally the cone CA_1. Let q:X\to Y denote the quotient map. It does seem to me like this space can be embedded in \mathbb{R}^3 although the pattern used to create Figure 1 might need to be modified a little.

Let’s make some observations about this space. Let y_0 be the image of x_0 in Y. This is obviously the point of interest since Y is locally contractible at all points in Y\backslash\{y_0\}. Moreover, this space is a Peano continuum and is therefore both locally 0-connected at all points. Therefore, we can focus on what’s going on with loops.

Let’s once again identify A_m with it’s image in Y. So \bigcup_{m\geq 1}A_m still forms an infinite earring space in Y (that is bolded in Figure 1). Let’s also write D_m for the image of the cone CA_m in Y. Note that D_m is not a cone but is a cone where it’s vertex has been attached to one of the points on the base circle. Most importantly, we have A_m=D_m\cap D_{m+1} for all m\geq 1.

Proposition 2: Y is simply connected.

Proof. By modifying standard cellular approximation methods, any given loop in Y based at y_0 is path-homotopic to a loop that has image in the earring space \bigcup_{m\geq 1}A_m. The idea here is to start with a loop and, for all m\geq 1, “push it off” a given point in the interior of D_m. This makes it easy to deformation retract the original loop to a loop L with image in \bigcup_{m\geq 1}A_m. But every loop L:[0,1]\to \bigcup_{m\geq 1}A_m lifts to a loop L':[0,1]\to X such that q\circ L'=L. Since X is contracible, L' is null-homotopic. It follows that L is null-homotopic. \square

Why isn’t Y locally 1-connected? Take any open neighborhood U of the basepoint y_0 which does not contain the largest cone-image D_1. Find the largest m such that D_m\nsubseteq U. Since A_m\subseteq D_{m+1}\subseteq U, the loop \alpha_m:[0,1]\to Y has image in U but it does not contract in U because the image of a contraction of \alpha_m would necessarily contain D_m. Thus U is not simply connected.

Why is Y an LC^1 space? Take any open neighborhood U of the basepoint y_0. Find the smallest m such that D_k\subseteq U for all k\geq m. Let V be a path-connected open neighborhood of y_0 such that V\subseteq U and such that A_k\nsubseteq V for all 1\leq k\leq m. Any loop in V based at y_0 can be deformed onto \bigcup_{k\geq m+1}A_k. But the argument used to prove Proposition 2 shows any such loop is null-homotopic in the subspace \bigcup_{k\geq m+1}D_k, which is a subset of U. Therefore, any loop in V contracts in U (even though it does not contract in V).

Theorem 3: There exists a 2-dimensional Peano continuum, which is LC^1 and locally 0-connected but not locally 1-connected.

So the equivalence of the global properties no longer holds when n\geq 1. I don’t really know if there is someone I should credit this Theorem to. No doubt it was known a long time ago so I’ll call it “folklore” until I learn otherwise.

Here are some brain teasers.

Question 4: Is Y contractible?

Question 5: Is there a LC^1 space that is not locally 1-connected at any point?

The answer to both of these questions should be “yes” but I haven’t thought too much about the details.

What about even higher dimensions? If you’re looking for an example of an LC^n space that is not locally n-connected you can mimic the above example by defining A_m to be an n-sphere instead of a circle. But the reason why the above example worked so well is because the paths \alpha_m are surjective – in order to contract \alpha_m in Y, you need to use all of D_m but doing so requires that you use all of the image of \alpha_{m-1}, which is A_{m-1}. Then your homotopy would have to use A_{m-2},A_{m-3},...,A_1, which is clearly a problem. To make this same thing happen in higher dimensions (using cones), we must realize that we don’t have a canoncial way to map the spine of a cone (which is an arc) onto a higher-dimensional sphere. How do we make \alpha_m:[0,1]\to A_m surjective like we did when n=1? If we choose \alpha_m to be a simple close curve, we won’t get you the kind of example we want. This is where we are thankful for space-filling curves! We can take \alpha_m:[0,1]\to A_m to be any continuous surjection, i.e. space-filling curve (such maps exist due to results in Continuum Theory). Once this choice is made, pretty much all of the arguments are the same for the n=1 case work the same. This leads us to the following extension of Theorem 3.

Theorem 6: For each n\geq 1, there exists a n-dimensional Peano continuum, which is LC^n and locally (n-1)-connected but not locally n-connected.

Posted in Algebraic Topology, Examples, General topology, Higher Homotopy groups, Homotopy theory, locally n-connected spaces, locally path connected, Peano continuum, Uncategorized | Tagged , , , , , , | Leave a comment

Local n-connectivity vs. the LC^n Property, Part I

Posted on October 8, 2023 by Jeremy Brazas

I’m going to take a little break from the topologized fundamental group series for a bit. That’s a long series and a distraction might be nice. Right now, I’m going to share a little about a property I’ve been running into pretty frequently. First, a standard definition in homotopy theory.

Definition 1: A space X is n-connected if \pi_k(X,x) is trivial for all 0\leq k\leq n (and some or any choice of basepoint), that is, if X is path connected (0-connected) and the first through n-th homotopy groups are trivial.

Definition 2: A space X is locally n-connected at x\in X if for every neighborhood U of x, there exists an n-connected neighborhood V of x such that V\subseteq U. A space is locally n-connected if it is locally n-connected at all of its points.

Lots of spaces are locally n-connected but when you’re proving general theorems it can be difficult or impossible to actually build n-connected open sets using general methods. There is a larger class of spaces that’s often more conducive to creating and proving general results. This is the LC^n property. The fact that so many old/classic papers use this property without defining it makes me feel that it used to be more broadly used and understood. Certainly, it’s still used in metric topology and shape theory (and of course wild topology). Even though the terminology and notation are definitely established, whenever I do use it, I feel the need to define it. Personally, I feel that unless one already knows the difference, it’d be easy to see “LCn ” for the first time and assume that this means locally n-connected…you know..because of the letters.

Definition 3: A space X is LC^n at x\in X if for every neighborhood U of x, there exists a neighborhood V of x such that V\subseteq U and such that for every 0\leq k\leq n, every map f:S^k\to V is null-homotopic in U. A space X is LC^n if it is LC^n at all of its points.

The key difference between the two definitions is that in the locally n-connected definition, V is chosen so that a map f:S^k\to V from a k-spheres (0\leq k\leq n) actually can be contracted in V. In the LC^n definition, V is chosen so that all maps f:S^k\to V contract within the slightly larger open set U. Certainly, we have:

locally n-connected at x \Rightarrow LC^n at x

Examples illustrating the difference between these properties can take us to wild places so let’s work some out in low dimensions. We’ll sort out the 0-dimensional case in the remainder of this post and discuss higher dimensions in the next post.

Local 0-connectivity and LC^0

A space X being locally 0-connected at a point x\in X means being locally path-connected. In this case, X has a basis of path-connected open sets at x. In contrast, LC^0 means that X is “relatively locally path-connected.” Given any small neighborhood U, we can find a smaller neighborhood V so that a map f:S^0\to V (where S^0=\{-1,1\}) extends to a path [-1,1]\to U. In short, any two points in V can be connected by a path in U but maybe not by a path in V. We could be worried that this path in U is too big. But remember that U was already an arbitrarily small neighborhood so it is still relatively small when compared with the neighborhood V. See below for an image of the two cases.

If a space is locally 0-connected at a point, V may be chosen so any map f from {-1,1} to V can be extended to a path within V (top). If the space is LC0 at a point, one may chose V but only be able to extend f to a path within U (bottom).

Because the LC^0 property is a relative-size-type property, we have to stretch our brains a bit when searching for an example that distinguishes the two properties. Here’s an example of a space with a point where you have the LC^0 property but not locally path connectivity.

Example 4: Consider the planar set illustrated below.

A compact planar set which is LC0 but not locally 0-connected at its rightmost point. Note that it does fail to be LC0 at all points in (0,1)\times \{0\}.

Let’s say we embed this space in the plane so the bottom arc is the closed unit interval on the x-axis. This space is locally path-connected at all points with positive y-component. It’s clearly not locally path connected at any point in [0,1)\times\{0\} but it’s less obvious what is happening at x=(1,0) because the fans are shrinking in diameter.

First, let’s show that this space is not locally 0-connected at the rightmost point x: Any neighborhood V of x contains one of the fan-vertices (1-2^{-n},0) for some smallest n. This is going to have to contain some of the points (1-2^{-n},y) for y>0 but there’s no way to connect x to such a point with a path in V because that path would need to pass through (1-2^{-(n-1)},0). One can deduce from this observation that any path-connected neighborhood of x must contain the entire bottom arc. So X is not locally path connected at x.

On the other hand, X is LC^0 at x. A basic neighborhood of x is of the form U_n=X\cap (1-2^{-n},1]\times \mathbb{R}, n\geq 1. Given n\geq 1, the set V=U_{n+2} is not path connected. However, any two points in V can be connected by a path in U_n because you’re allowed to move left just far enough to get to the nearest fan-vertex on the x-axis.

The above space is a compact, one-dimenisonal, planar metric space. However, it’s not a dendrite because it’s not locally connected (or even LC^0) at several points. Actually, in Example 4, we were only able to verify that X is LC^0 but not locally 0-connected at x because there are points besides x where X is not LC^0. Another way to look at it is this – in dimension n=0, the pointwise properties are not equivalent but the global properties are. This is formalized in the next proposition.

Proposition 5: A space X is LC^0 if and only if it is locally 0-connected.

Proof. One direction is clear. Suppose X is LC^0 at all of its points. Let x\in X and U be an open neighborhood of x. It suffices to show that the path component C of x in U is open in X. Let y\in C, then by assumption, we can find a neighborhood V of y such that V\subseteq U and such that every point v\in V can be connected to y by a path in U. But since C is the path component of y in U, we have V\subseteq C. This proves that C is open. \square

Sequentially 0-connected spaces

For first countable spaces, it’s possible to characterize the pointwise LC^0 property using the following definition.

Definition 6: A space X is sequentially 0-connected at x\in X if for every convergent sequence \{x_n\}\to x, there exists a path \alpha:[0,1]\to X such that \alpha((n-1)/n)=x_n and \alpha(1)=x. We say X is sequentially 0-connected if it is sequentially 0-connected at all of its points.

I’ve been finding sequential properties like this really useful lately and there are several equivalent ways to define this property. Basically this property means that every convergent sequence extends to a continuous path. The sequentially 0-connected property is the sequential analogue of the LC^0 property. I’ll justify this claim in the next theorem. In papers, I have claimed this proof is “elementary” but I don’t know of a reference for it. It will make me feel better if the proof is here.

Theorem 7: Suppose X is first countable at x\in X. Then X is LC^0 at x if and only if X is sequentially 0-connected at x.

Proof. For both directions, let U_1\supseteq U_2\supseteq \cdots be a neighborhood base at x. First, suppose X is LC^0 at x and let \{x_m\}\to x be a convergent sequence. Find a neighborhood V_n of x such that V_n\subseteq U_n and such that any two points in X can be connected by a path in U_n. Let n_m=\min\{n\mid x_m\in V_n\}. Since \{x_m\}\to x, it must be that \{n_m\}\to \infty. Let \beta_m:[0,1]\to U_n from x to x_m. Then every neighborhood of x contains \beta_m([0,1]) for all but finitely many m. This means the infinite path-concatenation \alpha=(\beta_{1}^{-1}\beta_2)(\beta_{2}^{-1}\beta_3)(\beta_{3}^{-1}\beta_4)\cdots is well-defined, continuous, maps (m-1)/m to x_m and 1 to 0.

For the other direction, suppose X is not LC^0 at x. Then there exists open neighborhood W of x such that for every neighborhood V of x in W, there exists two points in V that cannot be connected by a path in W. In particular, there must be some point v\in V such that x and v cannot be connected by a path in W (if this were not the case, the previous sentence would be violated using path concatenation). Let’s assume that we have our neighborhood base W\supseteq U_1\supseteq U_2\supseteq \cdots. Then for every n, there exists x_n\in U_n such that there is no path in W from x to x_n. Note that \{x_n\}\to x since we are choosing points from a neighborhood base. Also, it is not possible to find a continuous path \alpha:[0,1]\to X with \alpha((n-1)/n)=x_n and \alpha(1)=x. Indeed, such a path would have \alpha([(n-1)/n,1])\subseteq W for some n and thus restrict to a path \alpha|_{[(n-1)/n,1]}:[(n-1)/n,1]\to W from x_n to x; a contradiction. Since the convergent sequence \{x_n\}\to x cannot be extended to a path, we conclude that X cannot be sequentially 0-connected at x. \square

Since the space in Example 4 is a metric space, that example is sequentially 0-connected but not locally 0-connected. The next example is not a metric space but is more extreme. It’s topology is so large and fine that the space is sequentially 0-connected but not LC^0.

Example 8: Let O=\omega_1+1 be the first compact uncountable ordinal (with maximal element y_0=\omega_1. We give O the topology generated by singletons \{\lambda\}, \lambda<\omega_1 and the cofinal sets (\lambda,\omega_1]=\{\kappa\mid \kappa>\lambda\}. The key thing to know about O is that there is no sequence of countable ordinals \lambda_1<\lambda_2<\lambda_3<\cdots that converges to the maximal point y_0. For each \lambda<\omega_1, attach an arc A_{\lambda}=[0,1] to O by identifying 0\sim x_0 and 1\sim \lambda. Let Y be the resulting space with the weak topology with respect to the subspaces O and A_{\lambda}, \lambda<\omega_1. Any sequence in Y\backslash \{y_0\} that converges to y_0 eventually lies in A_{\lambda} for some fixed \lambda. All such sequences extend to a path and it follows that Y is sequentially 0-connected at y_0 (and thus at all of its points).

However, this space Y is not LC^0 at y_0. Let U be the union of the images of all copies of [0,1/3)\cup (2/3,1]\subset A_{\lambda} in Y. Any open neighborhood V of y_0 in U will contain elements of O but will not contain any arc A_{\lambda}. Thus there is no way to connect y_0 to the elements of O by a path in V or U. We conclude that Y is not LC^0 at y_0.

The space in Example 8 is constructed by attaching arcs to the first compact uncountable ordinal (but with a modified topology).

Example 9: There is an example of a locally path-connected space, which is not sequentially 0-connected. This rather remarkable example using the Axioms of Choice is constructed by Taras Banakh in an answer to this MO question of mine. It is even \Delta-generated!

Examples 4,8, and 9 show that the pointwise-sequentially 0-connected property is comparable to neither the locally 0-connected property nor the LC^0 property.

We’ll deal with the difference between local 1-connectivity and LC^1 in the next post.

Posted in Dendrite, first uncountable ordinal, Higher Homotopy groups, locally path connected, one-dimensional spaces, path components | Tagged , , , , , | 1 Comment