Activity for ronno
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Edit | Post #288097 |
Post edited: |
— | 5 months ago |
| Edit | Post #288097 | Initial revision | — | over 1 year ago |
| Answer | — |
A: Does this generalization of path-connectedness also cover general connectedness? 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-$\k... (more) |
— | over 1 year ago |
| Edit | Post #288096 | Initial revision | — | over 1 year ago |
| Answer | — |
A: Universal property of quotient spaces The issue is that for $g$ to be continuous it is not enough for $f$ to be continuous at $x$. It is not even enough for $f$ to be continuous at every $y \in [x]$, for example consider $f : \mathbf{R}{>0} \to \{0,1\}$ which takes $(n- 1/n, n + 1/n)$ to $1$ and every other point to $0$ and the equivalen... (more) |
— | over 1 year ago |