Are there sets that are not partitionable in certain ways?

[quasar987]

Are there sets that are not partitionable in certain ways? For exemple, can I partition $\mathbb{R}$ into a collection of singletons?

Can I partition $\mathbb{R}^2$ into a collection of lines of slope 2?

If so, how would you write each of those partitions?

Thx.

 

[Muzza]

For the first one: for every $$x \in \mathbb{R}$$ let $$U_x = \{x\}$$. Then $$\cup_{x \in \mathbb{R}} U_x = \mathbb{R}$$ and obviously all $$U_x$$ are disjoint from one another, and are singletons.

For the second: for every $$a \in \mathbb{R}$$ let $$U_a = \{(x, y) | y = 2x + a\}$$ (I wish I knew LaTeX better).

Then take $$(x_0, y_0) \in \mathbb{R}^2$$ and notice that $$(x_0, y_0) \in U_{y_0 - 2x_0}$$. Thus $$\cup_{a \in \mathbb{R}} U_a = \mathbb{R}^2$$. A simple calculation will also reveal that $$U_a$$ and $$U_b$$ are either equal or disjoint (for any real a, b).