Math and stuff

Proposition Show that every convex subset of $\mathbb{R}^n$ is simply connected. Conclude that $\mathbb{R}^n$ itself is simply connected. Solution Let $X$ be a convex subset of $\mathbb{R}^n$. Let $p \in X$. Let $[f] \in \pi_1(X, p)$. Then the straight-line homotopy $F(s, t) = f(s) + t(c_p(s) - f(s))$ shows that...

Proposition Let $X$ be any set whatsoever, and let $\mathscr{T} = \mathscr{P}(X)$ (the power set of $X$, which is the set of all subsets of $X$), so every subset of $X$ is open. This is called the discrete topology on $X$, and $(X, \mathscr{T})$ is called a discrete space. Let...

Proposition Let $(X_1, \mathscr{T}_1), \cdots, (X_n, \mathscr{T}_n)$ be topological spaces. Let $\mathscr{B} = \mathscr{T}_1 \times \cdots \mathscr{T}_n$. Prove that $\mathscr{B}$ is a basis for a topology. Solution $X_1 \times \cdots \times X_n \in \mathscr{B}$, so $\bigcup_{B \in \mathscr{B}} B = X_1 \times \cdots \times X_n$. Let $U_1 \times \cdots \times...

Proposition Let $X$ be a Hausdorff space, let $A \subset X$, and let $A’$ denote the set of limit points of $A$. Show that $A’$ is closed in $X$. Solution Let $x \in X \setminus A’$. Then $x$ is not a limit point of $A$. This implies that $x$ has...

Proposition Let $f: X \rightarrow Y$ be a continuous map between topological spaces, and let $\mathcal{B}$ be a basis for the topology of $X$. Let $f(\mathcal{B})$ denote the collection $\{ f(B) : B \in \mathcal{B} \}$ of subsets of $Y$. Show that $f(\mathcal{B})$ is a basis for the topology of...