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Stream: theory: algebraic topology

Topic: Delta generated spaces

Patrick Nicodemus (Aug 08 2022 at 17:30):

While discussing with a friend we realized that the Delta generated spaces are actually generated by paths.
I find this very surprising.
https://ncatlab.org/nlab/show/Delta-generated+topological+space
I was going to add the proof to this page but because of the current state of the nlab I don't think it's editable so i'm posting it here

Patrick Nicodemus (Aug 08 2022 at 17:32):

Let $\Pi$ be the functor $\mathbf{Top}\to\mathbf{Top}$ which associates to each space $X$ the finest topology such that every path in $X$ lifts to a path in $\Pi(X)$.

Patrick Nicodemus (Aug 08 2022 at 17:33):

Prop - Suppose $X$ is first countable and locally path-connected. Then $\Pi(X)$ is homeomorphic to $X$.

Patrick Nicodemus (Aug 08 2022 at 17:33):

Proof:

Patrick Nicodemus (Aug 08 2022 at 18:03):

Let $x$ be a point in $X$. We know already that the map $\Pi(X)\to X$ is continuous. Let $W$ be a neighborhood of $x$ in the $\Pi(X)$ topology; we will show that $W$ is a neighborhood of $x$ in the original topology.

Without loss of generality we can choose a small path connected open neighborhood of $x$ and restrict to considering this subspace. Therefore wlog suppose $X$ is path connected.

First-countable spaces are, in particular, sequential, so it suffices to show that every sequence $\{ x_n \}$ approaching $x$ in the original topology is eventually in $W$. Introduce such a sequence. Let $\{U_k\}$ be a countable neighborhood basis of $x$; without loss of generality suppose the sequence is descending. Let $\{V_k\}$ be a sequence of path-connected open neighborhoods of $x$ with $V_k\subset U_k$.

Without loss of generality assume that all the $x_n$ live in $V_0$, discarding from the sequence the finitely many that do not.

Define a function $T : \mathbb{N}\to\mathbb{N}$, where we let $T(k)$ be the greatest natural number such that $x_i \in V_{T(k)}$ for all $i \geq k$ (or $k$ if there is no upper bound.) $T(k)$ is monotonically increasing and grows to infinity.

For each natural number $k$ choose a path $\sigma_k$ from $x_k$ to $x_{k+1}$ in $V_{T(k)}$. Assemble these sequences into a single path $\sigma: I\to X$ which is defined by $\sigma_k$ on the interval $[1 - 2^{-k}, 1- 2^{-(k+1)}]$, and define $\sigma(1) = x$. It is obvious that $\sigma$ is continuous at all points less than $1$. To prove it is continuous at $1$, it suffices to prove that each $\sigma^{-1}(U_n)$ is a neighborhood of $1$. This holds because eventually all the segments that compose the path $\sigma$ lie in $V_k$ for some $k> n$, and $V_k \subset U_n$ for $k > n$$.

Thus $\sigma$ is continuous. Since $W$ is a neighborhood in the $\Pi(X)$ topology, there exists some $\epsilon$ such that $\sigma( (1-\epsilon, 1])\subset W$. But by the construction of $\sigma$ this means that all but finitely many points $x_n$ are in $W$, and since $X$ is sequential, $W$ is a neighborhood of $x$ in $X$. This concludes the proof.

Patrick Nicodemus (Aug 08 2022 at 18:07):

Applying this proposition, simplices are obviously first countable and locally path connected, thus $\Pi(\Delta^n)\cong \Delta^n$ for all $n$.

It follows that if $f : \Delta^n\to X$ is continuous, then $\Pi(f) : \Pi(\Delta^n) \to \Pi(X)$ can be composed with this homeomorphism to give a continuous map $\Delta^n\cong \Pi(\Delta^n)\to \Pi(X)$. So every singular $n$-simplex factors through the $\Pi$-topology, i.e. $\Pi$ is right adjoint to the forgetful functor from $\Delta$ generated spaces to all spaces.

Patrick Nicodemus (Aug 08 2022 at 18:08):

It runs badly counter to my intuition. Somehow one-dimensional paths should not be able to access or reconstruct higher dimensional data... but I think the proof is correct, anyway.

David Michael Roberts (Aug 09 2022 at 10:03):

The nLab is editable! That got sorted a little while back, the migration to the new software that was holding up being able to edit was canned. Please also pop in an edit summary if and when you do put in in there, and check the nForum for subsequent discussion

David Michael Roberts (Aug 09 2022 at 10:13):

Isn't the space $\Pi(X)$ the same underlying set as $X$ plus the final topology coming from all continuous functions $I\to X$? That is, we can write $\Pi(X)$ as a quotient $\coprod_{X^I} I \twoheadrightarrow \Pi(X)$. I'm wondering if there is anything that could be done with this quotient map, together with the one exhibiting $X$ itself as a quotient of a coproduct of topological simplices (of all dimensions), which give a commuting square. Maybe that's too formal.

David Michael Roberts (Aug 09 2022 at 10:16):

I like your idea, though. It's reminiscent of [[Boman's theorem]], which is very close to being (if not the same as) the analogous statement in the category of smooth manifolds, using $\mathbb{R}$ instead of $[0,1]$.

Patrick Nicodemus (Aug 09 2022 at 11:21):

David Michael Roberts said:

Isn't the space $\Pi(X)$ the same underlying set as $X$ plus the final topology coming from all continuous functions $I\to X$? That is, we can write $\Pi(X)$ as a quotient $\coprod_{X^I} I \twoheadrightarrow \Pi(X)$. I'm wondering if there is anything that could be done with this quotient map, together with the one exhibiting $X$ itself as a quotient of a coproduct of topological simplices (of all dimensions), which give a commuting square. Maybe that's too formal.

Yes, that's the space I'm talking about.

Zhen Lin Low (Aug 09 2022 at 11:59):

Hmmm. For comparison, is it surprising that the category of sequential spaces is generated by the "free convergent sequence"? I think this is a similar phenomenon. Your argument seems to show that, for a first-countable locally path connected space, knowing about the paths also tells you about the convergent sequences, so such spaces are also "path-generated". Since the standard simplices are first-countable locally path connected, this implies Δ-spaces are "path-generated".

Patrick Nicodemus (Aug 09 2022 at 12:33):

I see, that's a good observation

Reid Barton (Aug 09 2022 at 12:35):

There are surjective continuous maps $\Delta^1 \to \Delta^n$ for any $n$ (space-filling curves), which give an "explicit" description of $\Delta^n$ as a quotient of $\Delta^1$.

Zhen Lin Low (Aug 09 2022 at 12:42):

I was wondering about that whether the existence of space filling curves has anything to do with it, but I don't see how...

Tim Campion (Aug 15 2022 at 17:47):

Clark Barwick always refers to $\Delta$-generated spaces as "numerically-generated spaces", precisely because they are generated by $\mathbb{R}$.