Comments on Does this generalization of path-connectedness also cover general connectedness?
Parent
Does this generalization of path-connectedness also cover general connectedness?
I've got the following idea to generalise path-connectedness:
Given a topological space $P$ and a subspace $S$, a space $X$ is $(P,S)$-connected if every continuous function $f:S\to X$ can be extended to a continuous function $g:P\to X$.
The usual path-connectedness is obtained using $P=[0,1]$ with the usual topology and $S=\{0,1\}$. Since every function $f:\{0,1\}\to X$ is continuous, $f$ just picks two arbitrary points, and then $g$ is a path between those points.
Using the closed unit disc as $P$ and the unit circle as $S$ one also obtains whether all path-connected components are simply connected.
Now I wonder if there is also a choice of $P$ and $S$ that recovers general connectedness, that is the nonexistence of clopen sets other than the empty set and the full space.
Post
The following users marked this post as Works for me:
| User | Comment | Date |
|---|---|---|
| celtschk | (no comment) | Jun 8, 2024 at 09:03 |
This is not possible. Suppose there is such a $(P, S)$. Considering continuous maps to $\{0, 1\}$, there must be a clopen set $C \subset S$ such that there is no clopen $D \subset P$ with $C = D \cap S$.
Now let $\kappa < \lambda$ be infinite cardinalities bigger than $|P|$ and consider the co-$\kappa$ topology on a set $X$ of size $\lambda$. Then $X$ is connected but subsets of size $\le \kappa$ are discrete. Now consider $x_1 \ne x_2 \in X$ and the continuous map $f : S \to X$ given by $f(C) = x_1$, $f(S \setminus C) = x_2$. This has no continuous extension $g : P \to X$ since $x_1$ would be clopen in $g(P)$ and $D = g^{-1}(x_1)$ would satisfy $D \cap S = C$.
In short, $(P, S)$-connectedness can at best detect "connectedness at size $\le |P|$".
1 comment thread
0 comment threads