# Fundamental Groups

latentcorpse
I need to calculate the fundamental group of the following spaces:

$X_1 = \{ (x,y,z) \in \mathbb{R}^3 | x>0 \}$
$X_2 = \{ (x,y,z) \in \mathbb{R}^3 | x \neq 0 \}$
$X_3 = \{ (x,y,z) \in \mathbb{R}^3 | (x,y,z) \neq (0,0,0) \}$
$X_4 = \mathbb{R}^3 \backslash \{ (x,y,z) \in \mathbb{R}^3 | x=0,y=0, 0 \leqslant z \leqslant 1 \}$
$X_5 = \mathbb{R}^3 \backslash \{ (x,y,z) \in \mathbb{R}^3 | x=0, 0 \leqslant y \leqslant 1 \}$

I fundamentally do not understand what a fundamental group is of how to calculate it. I have read the notes on this but they are so so abstract.

e(ho0n3
What is the definition of a fundamental group?

latentcorpse
ok. i think i have answers for the first 3 that im happy with (ive discussed this with a coursemate)

the 4th one i believe the fundamental group is trivial as any path can be contracted to a point. is this correct?

and the 5th one is R£ with a "sheet" removed is was going to say that we can pull the space in around the sheet making a rectangle taht can then be deformed into a circle. this means the fundamental group of X5 is isomoprhic to that of S1 i.e. it is $\mathbb{Z}$. is this correct?

thanks.

latentcorpse
What is the definition of a fundamental group?

thanks. could you take a look at my above post please?