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9h
comment Example of continuous function over $\mathbb R^n$
Thanks. If I consider a function $f:X \to Y$, for which type of $f$ so that $f^{-1}(Y)=X$?
10h
comment Example of continuous function over $\mathbb R^n$
$f(t)=ty+(1-t)x$ is the special case that I think $f^{-1}(\mathbb R^n)=[0,1]$ would hold. In general, do all continuous functions $f$ work?
11h
comment Example of continuous function over $\mathbb R^n$
By definition the inverse image $f^{-1}(V) = \{x \in X \; | \; f(x) \in V \}$, can we deduce that $f^{-1}(\mathbb R^n) = [0,1]$? For which type of $f$ that equality can be occurred?
11h
revised How to understand Weyl chambers?
added 10 characters in body
11h
asked How to understand Weyl chambers?
13h
comment Example of continuous function over $\mathbb R^n$
And does your proof work for the Euclidean space instead of $\mathbb R^n$?
13h
comment Example of continuous function over $\mathbb R^n$
Thank you, I got it. By that definition, $[0,1]$ is also a closed set on itself, isn't it? I have one more question, is it true that $f^{-1}(\mathbb R^n) =[0,1]?$ for every function $f$?
1d
comment Example of continuous function over $\mathbb R^n$
Thank you. What is the topology in $[0,1]$?
1d
comment Example of continuous function over $\mathbb R^n$
For the case $x=y \in V$, why $[0,1]$ is open in $[0,1]$?
1d
comment Path-Connected implies Connected without knowing that [0,1] is connected
Why do you know that $[0,1]=\gamma^{-1}(X)?$
1d
comment Show that any convex subset of $R^k$ is connected
Anyone can explan why $P \cap U$ is open?
May
20
comment Intersection of positive half-spaces
Thank you Daniel Fischer. This is a very nice solution!
May
20
accepted Intersection of positive half-spaces
May
20
asked Intersection of positive half-spaces
May
20
comment Intersection of “positive” open half-spaces
Dear @copper.hat. At the last line of your proof, did you mean that $P=\phi^{-1}(A)?$
Apr
23
awarded  Popular Question
Apr
21
comment For $n \ge 2$ , does every linear operator on $\mathbb R^n$ has an invariant subspace of dimension $2$ ?
Thank you. I get it.
Apr
20
comment For $n \ge 2$ , does every linear operator on $\mathbb R^n$ has an invariant subspace of dimension $2$ ?
@EwanDelanoy. I have two more quick questions. First, what did you mean about the notation $f$ in the last paragraph? Did it mean the transformation $T$? Second, why $\mathbb R^n=\text{Ker}((f−\lambda id)^{s_1})$? Please confirm.
Apr
18
comment Determinant of a matrix with specific main diagonal
Thanks Chappers for your detailed answer. I actually found the same to your last expression. The result I used at the beginning was a modification for that. It's good to hear I ended up with the correct answer.
Apr
18
accepted Determinant of a matrix with specific main diagonal