Answer to Question #133629 in Differential Geometry | Topology for PRATHIBHA ROSE C S

Question

Prove
1. If X is connected, then every quotient space of X is connect .
2. If X is compact ,then every quotient of X is compact.

Answer 1

Let $Y$ be a quotient space of $X$. For every $x\in X$ denote the equivalence class of $x$ by $[x]$. Let $f\colon X\to Y$ be the mapping from $X$ to $Y$, where $f(x)=[x]$ for every $x\in X$. Then $f$ is continuous by definition of a quotient space.

1)Let $X$ be connected. Then $Y=f(X)$ is connected as continuous image of a connected set.

2)Let $X$ be compact. Then $Y=f(X)$ is compact as continuous image of a compact set.