pi-base
Properties related to arc connectedness
In this comment @marswill suggested introducing the notion of "totally arc disconnected", as it would allow to unify some theorems related to biconnected. After giving it some thought, I think the proposal makes sense.
Among the properties related to path connectedness, the only one that does not have a corresponding one for arc is P46: totally path disconnected = the only paths in $X$ are constant = all path connected components are singletons.
The new property would mean: all arcs in $X$ are constant (where arc = injective continuous map $f:[0,1]\to X$, plus the trivial constant path, which is considered an arc for convenience so that a point is always joined to itself by an arc). So possible names for this could be:
- "has no arcs"
- "has no nontrivial arcs"
- "totally arc disconnected"
Looking in google scholar and google books, I did not see anything for "totally arc disconnected" and only two or three references for "totally arcwise disconnected" (but in one case not defined, and in another as a synonym for totally path disconnected). Still, I think I like "totally arc disconnected" best here, for symmetry with the path equivalent?
Other issue: what do we mean by an arc? We follow S&S that define it as an injective continuous map $f:[0,1]\to X$. But it seems many (most?) other sources (including Willard and Engelking) require the stronger condition that $f$ be a homeomorphic embedding. Should we add variant properties for that?
Comments?
Other issue: what do we mean by an arc? We follow S&S that define it as an injective continuous map $f\colon [0,1]\to X$. But it seems many (most?) other sources (including Willard and Engelking) require the stronger condition that f be a homeomorphic embedding. Should we add variant properties for that?
Comments?
I've been thinking about this as well. I also often see arc-connectedness defined in the stronger sense you mention, where points must be connected by arcs, defined as homeomorphic images of an interval.
While a huge plethora of definitions might not be ideal, I do get the impression that arcs are pretty well-established as homeomorphic images of intervals, so the current definition we use is at least a little problematic. So I think I'd recommend either replacing it with the stronger one, or adding the stronger one and adding whatever appropriate modifiers to each name ("arc connected" and "bijectively path connected", would be a possibility.)
I'm not really sure where I ultimately come down on this. I see the argument that it's a stretch to have two arc-connectedness definitions (and then do we have a local version of each one, and a total disconnected version of each one?). On the other hand one could argue that if S&S use the bijective definition maybe that's a good enough excuse to have both, given that they are the inspiration for this site, and the other stronger definition seems quite relevant.
Also, for what it's worth, I think the theorem $T_2$+path connected implies arc-connected (T240) can probably be relaxed to only require $US$, under our current definition (following the sketch in this mse post). I have not carefully confirmed this.
On the other hand, with the stronger definition involving homeomorphic images, it can be relaxed to only need Weak Hausdorff, though not further (exercise: $\mathbb R$ with topology of Euclidean dense open sets is not weak-Hausdorrf, is $k_2$-Hausdorff, and is bijectively path connected but not arc-connected in the stronger sense. Hint: $KIP$).
So there is definitely potential for the weak and strong definitions to be interesting and distinct.
If we are going to have both, I am thinking replacing arc connected with the stronger definition would be more expected for users.
But we should keep the weaker concept as well. "injectively path connected"?
Maybe @ccaruvana or @StevenClontz will have other suggestions.
We follow S&S that defines an arc as an injective continuous map from $[0,1]$ to $X$.
https://ncatlab.org/nlab/show/connected+space#arcconnectedness uses the same definition.
Also, for what it's worth, I think the theorem $T_2$+path connected implies arc-connected (T240) can probably be relaxed to only require $US$, under our current definition (following the sketch in this mse post). I have not carefully confirmed this.
That would be very interesting to see. I'll check out that post.
Also, for what it's worth, I think the theorem $T_2$+path connected implies arc-connected (T240) can probably be relaxed to only require $US$, under our current definition (following the sketch in this mse post). I have not carefully confirmed this.
That would be very interesting to see. I'll check out that post.
Actually I was being silly, while that proof probably can be carefully modified to only use $US$, it would be a long and cumbersome process to check all the details, since the post only sketches the argument even for the $T_2$ case. One can instead directly generalize the result from needing $T_2$ to only needing $US$ as follows:
If $(X,\tau)$ is $US$, then $(X, \tau_s)$ is $US$ and sequential, where $\tau_s$ is the family of sequentially open sets in $\tau$. Note that since $[0,1]$ is sequential, $f\colon [0,1]\to X$ is continuous with respect to $\tau$ iff it is continuous with respect to $\tau_s$, so we can reduce to the case that $X$ is $US$ and Sequential, in which case $X$ is strongly KC, hence weak Hausdorff. In particular, the result follows from the fact that in a Weak Hausdorff space, the continuous image of a compact Hausdorff space is compact and Hausdorff, hence the image of any continuous path contains an arc by the $T_2$ result.
That is a very nice argument.
To fix terminology temporarily, let's say:
- $X$ is inj path conn if any two distinct points can be joined by an injective path = pi-base's notion of arc connected.
- $X$ is strong arc conn if any two distinct points can be joined by an arc that is a homeomorphism onto its image = other commonly used notion of arc connected in various sources.
(aside first, before coming back to your proof)
I was looking at the really nice expository article https://arxiv.org/pdf/2207.07242.pdf by Jeremy Brazas (mentioned in his answer to https://math.stackexchange.com/questions/1522642) that provides a direct proof of [path conn + T2 ==> strong arc conn]. It uses a Zorn's lemma argument to gradually excise loops from the path until obtaining an injective path. What makes this argument work is the technical notion $X$ "permits loop deletion" on p. 3.
In a first stage he proves [path conn + permits loop deletion ==> inj path conn].
And it is clear that US spaces satisfy "permits loop deletion". So that gives: [path conn +US ==> inj path conn].
In a second stage, every injective path in a T2 space is a homeo onto its image. So [path conn + T2 ==> strong arc conn].
That is generalized in section 2 by introducing the definition:
- $X$ is $\Delta$-Hausdorff if the image of any path $[0,1]\to X$ is closed in $x$.
Both $\Delta$-Hausdorff and US are intermediate between weak Hausdorff and T1. And both imply "permits loop deletion". So [path conn + $\Delta$-Hausdorff ==> inj path conn]. He then shows that injective paths in a $\Delta$-Hausdorff space are also homeo onto their image. This gives [path conn + $\Delta$-Hausdorff ==> strong arc conn].
I think it would be interesting to see the relationship between $\Delta$-Hausdorff and US. (A totally path disconnected space is $\Delta$-Hausdorff, but need not be US. But maybe in a locally path connected space, $\Delta$-Hausdorff would imply US for example?)
(back to your argument) Does it show inj path connected, or strong arc connected?
I need to think to see I am not missing anything.
(back to your argument) Does it show inj path connected, or strong arc connected?
I need to think to see I am not missing anything.
It only shows inj path connected for $US$ spaces. It does this by showing strong arc connected for $(X,\tau_s)$, and then we get inj path connected for $(X,\tau)$ by virtue of the fact that continuous maps from the sequential space $[0,1]$ into $X$ with either topology are the same, since they are precisely the sequentially continuous maps into $(X,\tau)$.
(back to your argument) Does it show inj path connected, or strong arc connected? I need to think to see I am not missing anything.
It only shows inj path connected for US spaces. It does this by showing strong arc connected for (X,τs), and then we get inj path connected for (X,τ) by virtue of the fact that continuous maps from the sequential space [0,1] into X with either topology are the same, since they are precisely the sequentially continuous maps into (X,τ).
Also, we have counterexamples that show you can be path connected and $US$ (even $k_2$-Hausdorff), yet not be strong arc connected - example, just give $\mathbb R$ the topology consisting of Euclidean dense open sets. It's pretty easy to show in this topology, the compact sets are the same as the Euclidean compact sets, so it is $KIP+T_1$, hence $k_2$-Hausdorff.
In this example, the induced sequential topology is just the Euclidean topology again (not hard to show you have the same convergent sequences), so again the paths are just the same as paths into the space with Euclidean topology, so the image of any nonconstant path is interval, which cannot be Hausdorff (two Euclidean dense open sets will intersect in any interval, so every interval is hyperconnected), hence the space is not strongly arc connected (in fact it's strongly totally arc-disconnected).
So your argument is a nice and simple way to extend T240 to [path connected + US ==> inj path connected].
And this whole discussion shows the value of having both arc connected properties, I would say.
Chris and I discussed in the zoom meeting and we both like "injectively path connected" and "arc connected" as names.
So "Inj. path conn. + weak Hausdorff => arc connected" by definition. Maybe that should be the new version of T240 (after T709 is there)? Or we can weaken the assumption even further. I am not familiar with all those properties below $T_2$.
So "Inj. path conn. + weak Hausdorff => arc connected" by definition. Maybe that should be the new version of T240 (after T709 is there)? Or we can weaken the assumption even further. I am not familiar with all those properties below T 2 .
This comment gives a counterexample that weak Hausdorff can't be weaken to $k_2 T_2$.
So "Inj. path conn. + weak Hausdorff => arc connected" by definition. Maybe that should be the new version of T240 (after T709 is there)? Or we can weaken the assumption even further. I am not familiar with all those properties below T 2 .
Continuing discussion in #1260.