Quotient space and quotient map

Question

Let $X:= [-2,2]$ a topological space with the canonical topology and let $A:=[-1,1]\subset X$ and $f:X \rightarrow X/A$ the quotient map.

How can I find an explicit form of the quotient space $X/A$?

And is $X$ a compact space? (I only know that it is complete, but I don't know if it is also compact)

Have someone any suggestions please? Thanks!

Answer 1

The quotient of a compact space is always compact, as $q: X \to X / \sim$ is always continuous (where $q$ is the standard quotient map), and images of compact spaces are compact.

In your case the function $f: X=[-2,2] \to [-1,1]$ where $f(x) = x+1$ for $x\in [-2,-1]$, $f(x) = 0$ for $x \in [-1,1]$ and $f(x) = x-1$ for $x \in [1,2]$ is continuous and obeys $f(x) = f(y)$ iff $x \sim y$ in the equivalence relation determined by identifying $A$ to a point.

A standard theorem then implies that $X/A = X / \sim \simeq f[X] = [-1,1]$.