3 votes
Accepted

A weakly locally path connected space that is not locally path connected

"Weakly locally path-connected" is actually equivalent to "locally path-connected". To prove this, suppose $X$ is weakly locally path-connected, $x\in X$, and $U$ is an open ...
user avatar
3 votes

Commutator subgroup of connected group.

The argument is a bit more complicated than what you suggest. For each $k$, there is a continuous map $f:G^{2k}\to G$ sending $(g_1,g_1',\dots,g_k,g_k')$ to $[g_1,g_1']\dots[g_k,g_k']$. So for each $...
user avatar
2 votes

Equation $x^k=g$ in connected group.

Let $A$ be a connected compact abelian group. Its Pontryagin dual $A^{\vee}$ is then a torsion-free discrete abelian group (see e.g. this math.SE answer) which determines $A$ since $A$ must be the ...
user avatar
1 vote

Determine if the following subsets in $\mathbb R^4$ are connected, path-connected, compact

We have $A=\{(x_1,x_2,x_3, x_4)\ |\ x_1^2-x_2^2+x_3^2-x_4^2=1\}$. Then $B$ is defined by the following system of equations $$\begin{cases} x_1^2-x_2^2+x_3^2-x_4^2=1 \\ x_2^2+x_4^2=1 \end{cases}$$ ...
user avatar
  • 33.1k
1 vote
Accepted

Continuous surjective function from $[0,1]$ to $[0,1]^2$

$f$ is closed, since its domain is compact. Thus $f_{|f^{-1}(U)}$ is closed (see this). This means that the inverse $g:U\to f^{-1}(U)$ is continuous. Therefore so is its composition with the inclusion ...
user avatar
  • 33.1k
1 vote

Showing that there is a finite chain of sets connecting any two distinct points in a connected metric space

The idea is correct and nice. You can do it a bit faster. To show that the set $A$ is open it sufficies to observe that all points from $U_n$ belong to $A$ (which chain?) and it's a nbhood od $a$. The ...
user avatar
  • 4,220
1 vote
Accepted

The limit set of an unbounded path/ray

I think I can find a path in ${\mathbb R}^2$ such that $L=C\times [0, \infty )$, where $C$ is Cantor's set, and hence $L$ has uncountably many connected components. Just take a path that attempts to &...
user avatar
  • 17.6k

Only top scored, non community-wiki answers of a minimum length are eligible