Math and stuff

Proposition Show that the only Hausdorff topology on a finite set is the discrete topology. Solution Let $X = \{ x_1, \cdots, x_n \}$ be a finite Hausdorff space. For each $i \in \{ 2, \cdots, n \}$, $x_1, x_i$ have disjoint neighborhoods $U_i, V_i$. Then $U_2 \cap \cdots \cap...

Proposition Suppose $f: X \rightarrow Y$ is a homeomorphism and $U \subset X$ is an open subset. Show that $f(U)$ is open in $Y$ and the restriction $f\mid_U$ is a homeomorphism from $U$ to $f(U)$. Solution By this post, $f\mid_U$ is continuous. $f \mid_U: U \rightarrow f(U)$ is surjective. $f...

Notes Vectors and derivations Fields Star Functions Definitions Push forwards ($F_*$) Pull backs ($F^*$) Vector fields Covector fields $df$ Exterior differentiation Set of all smooth vector fields ($\mathfrak{X}(M)$) Tangent Space ($T_pM$) Tangent Bundle ($TM$) Cotangent space ($T_p^*M$) Cotangent bundle (\(T^*M\)) Local framing Local framing of a tangent bundle Local framing...

Proposition Let $(X_1, \mathcal{T}_1)$ and $(X_2, \mathcal{T}_2)$ be topological spaces and let $f: X_1 \rightarrow X_2$ be a bijective map. Show that $f$ is a homeomorphism if and only if $f(\mathcal{T}_1) = \mathcal{T}_2$ in the sense that $U \in \mathcal{T}_1$ if and only if $f(U) \in \mathcal{T}_2$. Solution Suppose $f$...

Proposition Suppose $f: X \rightarrow Y$ is a bijective continuous map. Show that the following are equivalent: $f$ is a homeomorphism. $f$ is open. $f$ is closed. Solution $1 \rightarrow 2$ $f$ is a homeomorphism, so it must be open. $2 \rightarrow 3$ Let $C \subset X$ be closed. Then...