Sphere with g handles without point $\simeq \vee_{n=1}^{2g} \mathbb{S}^1$ Consider the standard representation of sphere with $g$ handles as CW-complex ($4g$-gon), obviously if we remove some point $p$, we obtain deformation retraction on border and if we factor border, we obtain $\vee_{n=1}^{2g} \mathbb{S}^1$.
But it's very unrigorous and maybe i can use the CW-approximation(or HEP-theorem) theorem, to make it more rigorous?
 A: If you are familiar with quotient maps and quotient topologies and how to use them, and how they are related to CW complexes, this argument is easily made completely rigorous. Here's a brief outline.
Let $S$ be the surface. Let $P$ be a regular $4g$-gon on the plane centered on a point $p$. Let $f : P \to S$ be a characteristic map of the unique 2-cell of the given CW structure on $S$, such that the restriction of $f$ to each side of $P$ is a characteristic map of one of the 1-cells. The map $f$ is a quotient map. The restriction $f \bigm| \partial P$ is a quotient map onto the $1$-skeleton $f(\partial P)$, which is homeomorphic to a wedge of $2g$ circles.
The radial map $r : P-p \mapsto \partial P$ is a deformation retraction. Also, the composed map 
$$R : P-p \xrightarrow{r} \partial P \xrightarrow{f} f(\partial P) 
$$
has the property that for each point $x \in f(\partial P)$, the function $R$ is constant on the set $f^{-1}(x)$. It follows that the function $R$ induces a continuous function $S-f(p) \mapsto f(\partial P)$. Inducing functions in this manner is a key technical lemma which is used to make quotient map arguments rigorous.
The radial homotopy --- the one that is used to demonstrate that the radial map is a deformation retraction --- is also constant on each point pre-image of $f$, and therefore it also descends to a homotopy of the induced function $S-p \mapsto f(\partial P)$, which one uses to demonstrate that this induced function is a deformation retraction.
