Math and stuff

Proposition If $M$ is an $n$-dimensional manifold with boundary, then $\Int{M}$ is an open subset of $M$, which is itself an $n$-dimensional manifold without boundary. Solution The manifold interior is defined differently from the topological interior. Let $x \in \Int{M}$. Then $x$ has a neighborhood $U$ that homeomorphic to an...

Proposition A map between topological spaces is continuous if and only if the preimage of every closed subset is closed. Solution Let $f: X \rightarrow Y$ be given. For any subset $C \subset Y$ and for any $x \in X$, \[\begin{align*} x \in f^{-1}(Y \setminus C) &\iff f(x) \in Y...

Proposition Give an example of a topological embedding that is neither an open map nor a closed map. Solution Let $A = [0, 1)$. Then the inclusion map $i_A: A \rightarrow \mathbb{R}$ is a topological embedding as shown here. $A$ is both open and closed in $A$. However, $i_A(A) =...

Proposition A surjective topological embedding is a homeomorphism. Solution Let $f:X \rightarrow Y$ be a topological embedding. By definition, $f$ is a homeomorphism from $X$ to $f(X)$. Suppose $f$ is surjective. Then $f(X) = Y$. Therefore, $f$ is a homeomorphism from $X$ to $Y$.

Proposition Let $X$ be a topological space and let $S$ be a subspace of $X$. Show that the inclusion map $S \rightarrow X$ is a topological embedding. Solution Let $i_S$ denote the inclusion map. Injective? $i_S(a) = i_S(b) \implies a = b$. Continuous? Let $U \subset X$. Then $i_S^{-1}(U) =...