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This problem is from Kosniowski's book "A First Course in Algebraic Topology". I am going to follow the hint but it is not easy for me to find $C_A$ and $C_B$. Is it correct that $C_A$ and $C_B$ should be the torus minus interior of a disk? If so, I could not see how that space contractible to a point. Thank you.

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    $\begingroup$ Note that every manifold is locally Euclidean, i.e., each point of an $n$-manifold has an open neighborhood homeomorphic to $\Bbb R^n$. So, $C_A$ will be a small open subset of $A$ containing the point $x_0$ and homeomorphic to $\Bbb R^2$. $\endgroup$
    – Sumanta
    Commented Oct 15, 2021 at 8:17
  • $\begingroup$ @SumantaDas Ah great, thank you for your comment. Do you know some works that don't need manifold? I haven't learned manifold yet in my courses $\endgroup$ Commented Oct 15, 2021 at 9:06
  • $\begingroup$ Have you seen the construction of the torus as a quotient of $[0, 1] \times [0, 1]$ by identifying the vertical and horizontal boundaries? This should allow you to conclude the same fact. $\endgroup$
    – Rushy
    Commented Oct 15, 2021 at 9:38
  • $\begingroup$ @Rushy yes I have, thank you, I will try to convince my self $\endgroup$ Commented Oct 15, 2021 at 11:20

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If you take the torus as $\mathbb{T}:=S^1\times S^{1}$ and pick points $y_{1},y_{2}\in S^{1}$ such that $x_{0}\notin \{y_{1}\}\times S^{1}$ and $x_{0}\notin S^{1}\times \{y_{2}\}$ then $X:=(S^{1}-\{y_{1}\})\times (S^{1}-\{y_{2}\})$ is an open neighborhood of $x_{0}$ by definition of the product topology and is contractible because $X$ is homeomorphic to $\mathbb{R}^{2}$.

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  • $\begingroup$ thank you, I will try to convince myself $\endgroup$ Commented Oct 15, 2021 at 11:22

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