# Math4202 Topology II (Lecture 1)

## Topology of manifolds

### Fundamental groups

Use fundamental group as invariant for topological spaces up to homeomorphism (exists bijective and continuous map with continuous inverse) / homotopy equivalence.

Classifying two dimensional surfaces.

- Sphere
- Torus
- $\mathbb{R}P^2$

## Quotient spaces

Let $X$ be a topological space and $f:X\to Y$ is a

1. continuous
2. surjective map.
3. With the property that $U\subset Y$ is open if and only if $f^{-1}(U)$ is open in $X$.

Then we say $f$ is a quotient map and $Y$ is a quotient space.