Developing Examples for Basis $\mathcal{B}$ on Topology $\tau$ I put together my own example on the basis of topology. I wanted to know if it is a valid example displaying the properties of basis. Here is my example:
Example: Since $X\subset\mathbb{R}$, let $X=(0,1)$ and $x=0.3\in{X}$. From this, we know that a basis $\mathcal{B}$ is a collection of subsets of $X$ and $B$ is known as the basis elements of $\mathcal{B}$ which are the subsets of $X$. So, we can define two different subsets of $X$ denoted by $B_1$ and $B_2$:
$$B_1=(0.1,0.6),$$
$$B_2=(0.2,0.9).$$
From this, we can see that they both contain $x$ and are subsets of $X$. However, there is also an intersection. So,
$$B_1\cap{B}_2=(0.2,0.6).$$
From this, we can still see that $x=0.3\in{B}_1\cap{B}_2$. So, to satisfy the last condition of a basis, we need to define a $B_3$ that is a subset of $B_1\cap{B}_2$ in which $x\in{B}_3$ (e.g., $x\in{B}_3\subset{B}_1\cap{B}_2$). So we define $B_3$ as 
$$B_3=(0.25,0.4),$$
which is a subset of $B_1=(0.1,0.6)\cap{B}_2=(0.2,0.9)$ and contains $x=0.3$. 
Therefore, by the satisfied conditions of a basis, basis $\mathcal{B}$ is a basis of topology $\tau$ on set $X=(0,1)$.

So a few questions I wanted to get addressed:


*

*Is this example even valid and does it show an understanding of the definition of a basis? If you have any edits for me, please provide me with specific corrections. I want to be as clear in my example explanations as possible for a better understanding or recognition if I end up looking back at my notes in the future for a refresh on this topic for learning higher mathematics.

*Would this example work for showing an example for the definition of a "topology $\tau$ generated by $\mathcal{B}$? If not, can you provide me with an example? Here is the definition of "this" that was given: Let $\mathcal{B}$ be a basis for a topology on $X$. Then the collection of subsets $U\subset{X}$ such that 


$$\forall{x}\in{U}, \exists{B}\in\mathcal{B} \ \text{with} \ x\in{B}\subset{U}.$$


*If you have anymore examples, can you show me them? I would also like to see visuals if possible. I am trying to gain the best intuition possible to build my strong foundation.



Edit ($2$/$6$/$2015$): Here is a revised version of my example:
Example (REVISED): Let $X\subset\mathbb{R}$ and $x=0.3\in{X}$. Here, we can define two different subsets of $X$ denoted by $B_1$ and $B_2$:
$$B_1=(0.1,0.6),$$
$$B_2=(0.2,0.9).$$
From this, we can see that they both contain the point $0.3$ and are subsets of $X$. Hence we can also define an intersection between $B_1,B_2$:
$$B_1\cap{B}_2=(0.2,0.6).$$
From this, we can still see that $0.3\in{B}_1\cap{B}_2$. Hence, to satisfy the last condition of a basis, we need to define a $B_3$ that is a subset of $B_1\cap{B}_2$ in which $0.3\in{B}_3$ (e.g., $0.3\in{B}_3\subset{B}_1\cap{B}_2$). So we define $B_3$ to be 
$$B_3=(0.25,0.4),$$
which is a subset of $B_1=(0.1,0.6)\cap{B}_2=(0.2,0.9)$ and contains $x=0.3$. 
Therefore, $\mathcal{B}$ is a basis of topology $\tau$ on the set $x=(0,1)$.
 A: (i) You need $\mathcal B$ to cover $X=(0,1)$ for it to be a basis for a topology (by definition of basis). The sets $B_1,B_2,B_3$ you have defined do not cover the interval $X=(0,1)$, but if you make the left and right endpoints of $B_1$ and $B_2$, respectively, $0$ and $1$ you're ok here.
(ii) Also for $\mathcal B$ to be a basis, you need: For each $B',B''\in\mathcal B$ and $x\in B'\cap B''$ there exists $B'''\in \mathcal B$ such that $x\in B'''\subseteq B'\cap B''$. The $\mathcal B$ you have defined satisfies this property at $x=1/3$, that is, whenever $B',B''\in \mathcal B$ and $1/3\in B'\cap B''$, there exists $B'''\in \mathcal B$ such that $1/3\in B'''\subseteq B'\cap B''$. But what if we take $x=.5\in B_1\cap B_2$. Is there an element of your $\mathcal B$ which contains $.5$ and is contained in $B_1\cap B_2$? No. You can avoid this problem if you define $B_3=(0.2,0.6)$ $(=B_1 \cap B_2)$.
Now you have a basis for a topology on $X$. The topology it generates is actually just $\mathcal B$.  
Another simple example: Let $X=\{0,1,2,3,...\}$. Can you show that $$\mathcal B=\{X\setminus F:F\subseteq X \text{ is finite}\}$$ is a basis for a topology on $X$?
