To characterize equivalence classes of homeomorphism, we use topological invariants that are conserved under homeomorphisms. For example, the number of connected components in the space, the algebraic structure(group, ring), compactness, connectedness, or Hausdorff property.

• $y=x^{2}\nsim x^{2}-y^{2}=1$, due to the connect components
• $S^{1}\nsim \mathbb{R}$, $S^{1}$ is compact while $\mathbb{R}$ is not
• $S^{1}\nsim [-1,1]$, $[-1,1]$ is simply connected but $S^{1}$ is not. For more evidence, if we remove point $0$, $[-1,1]$ is unconnected and $S^{1}$ is still connected.
• An open disc $D^{2}=\{(x,y)\mid x^{2}+y^{2}<1\}$ $\sim \mathbb{R}^{2}$
• An closed disc with boundary $S^{1}$ corresponding to a point is isomorphic to $S^{2}$ or $\mathbb{R}^{2}\cup \infty$. If we add infinity to $\mathbb{R}^{2}$ we get a compact space $S^{2}$.
  We can relax the condition so that the continuous function $f:X_{1}\to X_{2},g:X_{2}\to X_{1}$ do not have inverses:

  If $f:X\to Y,g:Y\to X$ are continuous functions, then $X,Y$ are of the same homotopy type.

  For example,

• $f(x)=0,g(0)=\frac{1}{2}$ are two functions on $X=(0,1),Y=\{0\}$, so they are of the same homotopy type.
• $S^{1}$ and cylinder $S^{1}\times \mathbb{R}$ is of same homotopy type.

$\chi(T)=V-E+F$ where $V,E,F$ are vertices, edges, faces of a polyhedron homeomorphic to $T$.
Torus $\Sigma_{g}$( g handles) : $\chi(\Sigma_{g})=2-2g$
Torus $T^{2}=\Sigma_{1}$: $\chi(T^{2})=0$
Sphere: $\chi(S^{2})=2$(Euler's theorem)
Mobius strip $\chi=0$
Connected sum $\#$: $T^{2}\#T^{2}=\Sigma_{2},\underbrace{ T^{2}\#T^{2}\#\dots T^{2} }_{ g\text{ factors} }=\Sigma_{g}$
We can glue $4g$-edge polygon to a torus with $g$-genus:

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