Return to Question

Given a topological space $C$ and points $c_0, c1\in C$ we say that a topological space $(X,\tau)$ is $(C,c_0,c_1)$-connected if and only if for all $x,y\in X$ there is $f:C\to X$ continuous with $f(c_0) = x$ and $f(c_1)=y$.

(In this language, path-connectedness equals $([0,1],0,1)$-connectedness.)

What is an example of a space $C$ and points $c_0\neq c1\in C$ such that

1. $(C,c_0,c_1)$-connectedness implies path-connectedness, and
2. there is a space that is path-connected, but not $(C,c_0,c_1)$-connected

?

Given a topological space $C$ and points $c_0, c1\in C$ we say that a topological space $(X,\tau)$ is $(C,c_0,c_1)$-connected if and only if for all $x,y\in X$ there is $f:C\to X$ continuous with $f(c_0) = x$ and $f(c_1)=y$.

(In this language, path-connectedness equals $([0,1],0,1)$-connectedness.)

What is an example of a space $C$ and points $c_0\neq c1\in C$ such that

1. $(C,c_0,c_1)$-connectedness implies path-connectedness, and
2. for every infinite cardinal $\kappa$ there is a topology on $\tau$ on $\kappa$ such that $(\kappa,\tau)$ is path-connected, but not $(C,c_0,c_1)$-connected

?

Given a topological space $C$ and points $c_0, c1\in C$ we say that a topological space $(X,\tau)$ is $(C,c_0,c_1)$-connected if and only if for all $x,y\in X$ there is $f:C\to X$ continuous with $f(c_0) = x$ and $f(c_1)=y$.

(In this language, path-connectedness equals $([0,1],0,1)$-connectedness.)

What is an example of a space $C$ and points $c_0, c1\in C$ such that

1. $(C,c_0,c_1)$-connectedness implies path-connectedness, and
2. there is a space that is path-connected, but not $(C,c_0,c_1)$-connected

?

Given a topological space $C$ and points $c_0, c1\in C$ we say that a topological space $(X,\tau)$ is $(C,c_0,c_1)$-connected if and only if for all $x,y\in X$ there is $f:C\to X$ continuous with $f(c_0) = x$ and $f(c_1)=y$.

(In this language, path-connectedness equals $([0,1],0,1)$-connectedness.)

What is an example of a space $C$ and points $c_0\neq c1\in C$ such that

1. $(C,c_0,c_1)$-connectedness implies path-connectedness, and
2. there is a space that is path-connected, but not $(C,c_0,c_1)$-connected

?

Given a topological space $C$ and points $c_0, c1\in C$ we say that a topological space $(X,\tau)$ is $(C,c_0,c_1)$-connected if and only if for all $x,y\in X$ there is $f:C\to X$ continuous with $f(c_0) = x$ and $f(c_1)=y$.

(In this language, path-connectedness equals $([0,1],0,1)$-connectedness.)

What are examples of a space $C$ and points $c_0, c1\in C$ such that

1. $(C,c_0,c_1)$-connectedness implies path-connectedness, and
2. there is a space that is path-connected, but not $(C,c_0,c_1)$-connected

?

Given a topological space $C$ and points $c_0, c1\in C$ we say that a topological space $(X,\tau)$ is $(C,c_0,c_1)$-connected if and only if for all $x,y\in X$ there is $f:C\to X$ continuous with $f(c_0) = x$ and $f(c_1)=y$.

(In this language, path-connectedness equals $([0,1],0,1)$-connectedness.)

What is an example of a space $C$ and points $c_0, c1\in C$ such that

1. $(C,c_0,c_1)$-connectedness implies path-connectedness, and
2. there is a space that is path-connected, but not $(C,c_0,c_1)$-connected

?