Math and stuff

Proposition Let $\phi_t: X \rightarrow Y$ is a homotopy and $h$ is the path $\phi_t(x_0)$ formed by the images of a basepoint $x_0 \in X$. Then \(\phi_{0*} = \beta_h \phi_{1*}\). Solution Let $[f] \in \pi_1(X, x_0)$. We claim that \(\phi_{0*}([f]) = \beta_h(\phi_{1*}([f])\). Let $F(x, t) = \phi_t(x)$ be the associated...

Proposition $\mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^n$ for $n \ne 2$. Solution Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^n$ be a homeomorphism. Then the restriction $f_{\mathbb{R}^2 \setminus 0}$ is a homeomorphism from $\mathbb{R}^2 \setminus \{ 0 \}$ into $\mathbb{R}^n \setminus \{ f(0) \}$. $\mathbb{R}^n \setminus \{ f(0) \}$ is homeomorphic to $\mathbb{R}^n...

Proposition For any $n \in \mathbb{N}$, $S^n \setminus \{ (0, 0, \cdots, 0, 1) \}$ is homeomorphic to $\mathbb{R}^{n}$. Solution Let $p = (0, 0, \cdots, 0, 1)$. We define two functions: $f: S^n \setminus p \rightarrow \mathbb{R}^n$ such that $(x_1, \cdots, x_n, x_{n + 1}) \mapsto (x_1 / (1...

Proposition If a space $X$ is the union of collection of path-connected open sets $A_{\alpha}$ each containing the basepoint $x_0 \in X$ and if each intersection $A_{\alpha} \cap A_{\beta}$ is path-connected, then every loop in $X$ at $x_0$ is homotopic to a product of loops each of which is contained...

Proposition $\pi_1(S^n) = 0$ if $n \geq 2$. Solution Let $p = (0, 0, \cdots, 0, 1)$. Let $A_1 = S^n \setminus \{ p \}, A_2 = S^n \setminus \{ -p \}$. Then $A_1, A_2$ are both open and $A_1 \cup A_2 = S^n$. As shown in this post, $A_1,...