# The topology of $S^2 \setminus \{ \text{points} \}$

I am confusing about sphere minus points. [I mean $$S^2 \setminus \{ points \}$$]

First, , I understand $$S^2$$ $$\setminus$$ {North pole} homeomorphic $$\mathbb{R}^2$$. [I can understand this using sterographic projection]

Next, $$S^2$$ $$\setminus$$ {North pole, South pole} is a homeomorphic to Cylinder ($$S^1 \times \mathbb{R}$$), actually geometrically(I mean by drawing) I can understand this procedure. Of course I can explicitly find the homeomorphism. [for example, intuitively Why do we have $S^2-\{N,S\}\simeq S^1\times\mathbb{R}$? or constructing homeomorphism, How can I prove that a cylinder is diffeomorphic to a twice-punctured $n$-sphere? ]

Now I want further, As a next step I tried to understand three points case.

I edit question in a following way.

Since $$S^2 \setminus$$ {three points} is homotpoic to figure 8 (not homeomoprhic). I am concern with its homeomorphic shape.

Is there any name for $$S^2 \setminus$$ {three points}? I simply guess it is homeomorphic to $$\mathbb{R}^2 \setminus$$ {two points}, but can not draw its shape in my head.

$$S^2 \setminus$$ {more than three points} $$\simeq ?$$
• A sphere minus $3$ points is not homeomorphic to a figure $8$, but it is homotopy equivalent to that space. – Cheerful Parsnip Jun 12 at 15:40
• The sphere minus three points is homotopy equivalent to a figure eight. By the way, it doesn't matter which point you remove, or which two points. $S^2\setminus \{x_1,x_2,\dots,x_n\}$ choosing $n$ distinct points will return the same space, up to homomorphism, for any two ways to pick the $x_i.$ – Thomas Andrews Jun 12 at 15:46
• The sphere minus $n$ points, for $n>1$ is homotopy equivalent to $$\underbrace{S_1\vee S_1\vee\cdots\vee S_1}_{n-1\text{ times}}$$ – Thomas Andrews Jun 12 at 15:52