Math and stuff

Proposition Let $S \subset X$ and $\mathscr{T}_S = \{ S \cap U \mid U \in \mathscr{T} \}$. Then $\mathscr{T}_S$ is a topology on $S$. Solution $\emptyset$ is in $\mathscr{T}$, so $S \cap \emptyset = \emptyset \in \mathscr{T}_S$. $X$ is in $\mathscr{T}$, so $S \cap X = S \in \mathscr{T}_S$. Let...

Proposition Le $X$ be a set, and suppose \(\{ \mathscr{T}_{\alpha} \}_{\alpha \in A}\) is a collection of topologies on $X$. Show that the intersection $\mathscr{T} = \cap_{\alpha \in A} \mathscr{T}_{\alpha}$ is a topology on $X$. Solution Each $\mathscr{T}_{\alpha}$ contains $\emptyset$ and $X$ because it is a topology. Thus the intersection...

Proposition Let $Y$ be a topological space with the trivial topology. Show that every sequence in $Y$ converges to every point of $Y$. Solution Let $\{ y_n \}$ be a sequence in $Y$. Let $y \in Y$ be given. Let $U$ be a neighborhood of $Y$. Since $Y$ has the...

Proposition Let $F: I \times I \rightarrow X$ be a continuous map, and let $f, g, h$, and $k$ be the paths in $X$ defined by \[\begin{align*} f(s) &= F(s, 0); \\ g(s) &= F(1, s); \\ h(s) &= F(0, s); \\ k(s) &= F(s, 1). \end{align*}\] Then $f \cdot...

Proposition Let $X$ be a path-connected topological space, and let $p, q \in X$. Show that all paths from $p$ to $q$ give the same isomorphism of $\pi_1(X, p)$ with $\pi_1(X, q)$ if and only if $\pi_1(X, p)$ is abelian. Solution Suppose that all paths from $p$ to $q$ give...