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Some topology problems

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#1
futurus

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1. Let ~ be the equivalence relation (x_{1}, x_{2}) ~ (y_{1}, y_{2}) iff x_{2}=y_{2}, on R^{2}. Then R^{2}/~ is homeomorphic to R.

2. Let D be the decomposition of the plane into concentric circles about the origin. Prove that D is homeomorphic to {x :D R|x :D 0}.

3. Find a counterexample for:
"E :D X is disconnected iff there are disjoint open subsets H and K in X, each meeting E, such that E :) H :cup K."

Thank you,

#2
bangbang1412

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1. Let ~ be the equivalence relation (x_{1}, x_{2}) ~ (y_{1}, y_{2}) iff x_{2}=y_{2}, on R^{2}. Then R^{2}/~ is homeomorphic to R.

2. Let D be the decomposition of the plane into concentric circles about the origin. Prove that D is homeomorphic to {x image003.gif R|x image004.gif 0}.

3. Find a counterexample for:
"E image075.gif X is disconnected iff there are disjoint open subsets H and K in X, each meeting E, such that E image001.gif H hop.gif K."

Thank you,

$1)$ Easy , you can see that this equivalence just  shrink or deformation of $R^{2}$ on $R$ where you connect all point to a line .

$2)$ Latexx

$3)$ By definition 


Bài viết đã được chỉnh sửa nội dung bởi bangbang1412: 22-03-2017 - 19:38

$$[\Psi_f(\mathbb{1}_{X_{\eta}}) ] = \sum_{\varnothing \neq J} (-1)^{\left|J \right|-1} [\mathrm{M}_{X_{\sigma},c}^{\vee}(\widetilde{D}_J^{\circ} \times_k \mathbf{G}_{m,k}^{\left|J \right|-1})] \in K_0(\mathbf{SH}_{\mathfrak{M},ct}(X_{\sigma})).$$





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