Topology: An Inquiry-Based Approach

Prove that a subset $O$ of a metric space $X$ is open if and only if $O=\text{Int}(O)\text{.}$

Let $a,b\in \mathbb{R}$ with $a<b\text{.}$ Show that the set $(a,b]$ in $(\mathbb{R},d_E)$ is not an open set.

Let $S$ be a finite set of points in ${\mathbb{R}}^2\text{.}$ Is the set ${\mathbb{R}}^2\setminus A$ an open set in $({\mathbb{R}}^2,d_E)\text{?}$ Prove your answer.

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces and let $f:X\to Y$ be a function. Prove that $f$ is continuous if and only if $f^{-1}(\text{Int}(B))\subseteq \text{Int}(f^{-1}(B))$ for every subset $B$ of $Y\text{.}$

Let $(X,d)$ be a metric space and let $x_1$ and $x_2$ be distinct points in $X\text{.}$ Prove that there are open sets $O_1$ containing $x_1$ and $O_2$ containing $x_2$ such that $O_1\cap O_2=\mathrm{\varnothing }\text{.}$ (This shows that we can separate points in metric spaces with open sets. Separation properties are important in topology.)

Let $f:{\mathbb{R}}^2\to \mathbb{R}$ be defined by $f((x,y))=x\text{.}$ Assume that we use the max metric $d_M$ on ${\mathbb{R}}^2$ and the Euclidean metric $d_E$ on $\mathbb{R}\text{.}$ Use Theorem to determine if $f$ a continuous function. Prove your conjecture.