Continuity

due by Monday, Sep 13, 2021

This assignment is based on material in lectures 7 and 8.

1. Let $f: X\to Y$ be a function and $A, B\subset X$. Prove or disprove the following.

1. $f(A\cup B)\subset f(A)\cup f(B)$.
2. $f(A\cup B)\supset f(A)\cup f(B)$.
3. $f(A\cap B)\subset f(A)\cap f(B)$.
4. $f(A\cap B)\supset f(A)\cap f(B)$.
2. Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. A map $f: X\to Y$ is said to be Lipschitz if there exists a constant $K> 0$ such that $$\forall p, q\in X,\ d_Y(f(p), f(q)) \leq K d_X(p, q).$$ Prove or disprove the following.

1. Every Lipschitz map is continuous.
2. Every continuous map is Lipschitz.
3. Let $Y$ be a topological space such that for all topological spaces $X$, every map $f: X\to Y$ is continuous. Prove that $Y$ has the indiscrete topology.

4. Suppose $f: X\to Y$ is a map between topological spaces and $A\subset X$ is a subspace. Prove or disprove the following.

1. If the restriction $f\vert_A: A \to Y$ is continuous, then for all $x\in A$, $f$ is continuous at $x$.
2. If, for all $x\in A$, $f$ is continuous at $x$, the restriction $f\vert_A: A \to Y$ is continuous.
5. Let $X$ be a space with the discrete topology. Prove or disprove the following.

1. If $x\in X$, then any neighbourhood basis of $x$ contains the singleton set ${x}$.
2. Every point has a neighbourhood basis with just one open set.
6. Let $f, g: X\to \mathbb{R}$ be continuous functions from a topological space to the reals. Prove or disprove that the following must be continuous?

1. $\min(f, g)$.
2. $\max(f, g)$.
3. $\vert f\vert - \vert g\vert$.