Determine the components of the space. 
For each $p \in \mathbb{R}$ and $\epsilon > 0, \: (\epsilon \in \mathbb{R}).$
  let $$\beta_{\epsilon}(p)=\{x \in \mathbb{R}^2: |x-p| < \epsilon \:\:\text{and}\:\: x_2 \neq p_2\}\: \cup \: \{p\}$$
Here $x_2$ and $p_2$ are the second coordinates of $x$ and $p$, respectively.
  Show that this collection of subsets is a topology on $\mathbb{R}^2$ and determine the components of the resulting space.


The collection covers $\mathbb{R}^2$ since any point $p \in \beta_1(p)$.
To see that the collection is a base for a topology we have to check that a point in the intersection of two basic open sets is in another basic open set contained in both of them.
If $q$ is a point in the intersection which is not a “center point” of any of the two basic open sets, then there is a usual metric disk of radius $\epsilon$ around $q$ contained in one of the basic open set. This disk contains $\beta_{\epsilon}(q)$.

What do they mean by "determine the components of the space" and can anyone give me a hint or two on how to do that?
Thanks.

If my memory serves me right,' components' means maximal connected subspaces. A topological space $X$ is said to be connected if
$$A,B \subseteq X \text{ closed }: A \cup B = X, A \cap B = \emptyset \Rightarrow \{A,B\} = \{X,\emptyset\}$$
or equivalently,
$$A,B \subseteq X \text{ open }: A \cup B = X, A \cap B = \emptyset \Rightarrow \{A,B\} = \{X,\emptyset\}.$$
So basically you would have to look for subspaces of your given top. space that have this property. Since components are maximal subspaces with this property, you can just look for subsets which are open and closed simultaneously in the topology of the whole space.
