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latentcorpse

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For [itex] n \geq m \geq 0[/itex], V is an m dimensional subspace of [itex]\mathbb{R}^n[/itex] and [itex]X=\mathbb{R}^n \backslash V[/itex].

Let [itex]\pi_0 ( X ) = X / \sim[/itex] be the identification space of X where [itex]x_0 \sim x_1[/itex] if there exists a path [itex]\alpha : [0,1] \rightarrow X[/itex] with [itex]\alpha(0)=x_0, \alpha(1)=x_1[/itex].

I'm asked to find the no. of path components, [itex] | \pi_0 (X) |[/itex] (the answer varies with m and n).

i've had a go...

my understanding so far is that if m=0 then we remove a point from [itex]\mathbb{R}^n[/itex] but we can still create paths from every other point in [itex]\mathbb{R}^n[/itex] to every other point in [itex]\mathbb{R}^n[/itex] so [itex] | \pi_0 (X) | = 1[/itex] provided [itex]n>1[/itex]. if [itex]n=1[/itex] then [itex] |\pi_0 (X)|=2[/itex]. if n=0 then i guess the set of path components would empty as X is empty.

extrapolating this to higher dimensional m and n,

surely [itex]|\pi_0 (X)| = \begin{cases} 1 \text{ if } n > m+1 \\ 2 \text{ if } n=m+1 \\ 0 \text{ if } n=m \end{cases}[/itex]

what do you reckon?

Let [itex]\pi_0 ( X ) = X / \sim[/itex] be the identification space of X where [itex]x_0 \sim x_1[/itex] if there exists a path [itex]\alpha : [0,1] \rightarrow X[/itex] with [itex]\alpha(0)=x_0, \alpha(1)=x_1[/itex].

I'm asked to find the no. of path components, [itex] | \pi_0 (X) |[/itex] (the answer varies with m and n).

i've had a go...

my understanding so far is that if m=0 then we remove a point from [itex]\mathbb{R}^n[/itex] but we can still create paths from every other point in [itex]\mathbb{R}^n[/itex] to every other point in [itex]\mathbb{R}^n[/itex] so [itex] | \pi_0 (X) | = 1[/itex] provided [itex]n>1[/itex]. if [itex]n=1[/itex] then [itex] |\pi_0 (X)|=2[/itex]. if n=0 then i guess the set of path components would empty as X is empty.

extrapolating this to higher dimensional m and n,

surely [itex]|\pi_0 (X)| = \begin{cases} 1 \text{ if } n > m+1 \\ 2 \text{ if } n=m+1 \\ 0 \text{ if } n=m \end{cases}[/itex]

what do you reckon?

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