What the following statement exactly mean?

[lee.spi]

what the following statement exactly mean :"A topological space is a set M with a distinguished collection of subsets,to be called the open sets".and I don't understand the meaning of "distinguished collection ".it's better have some example.thanks

 

[micromass]

It means that in order to give a topological space, you need to give two pieces of information:

1) First, you need to give an underlying set $X$
2) Secondly, you need to give a set $\mathcal{T}$ which consist of subsets of $X$. This set $\mathcal{T}$ is called the topology. It can be completely arbitrary, but it does need to satisfy the axioms of a topological space.

For example,
1) I give you the set $X=\mathbb{R}$
2) I give the topology $\mathcal{T} = \{\emptyset,\mathbb{R}\}$.
These two pieces of data specify a topological space, called an indiscrete space.

Other example:
1) I give you the set $X=\{0,1\}$
2) I give the topology $\mathcal{T} = \{\emptyset,\{0,1\},\{0\}\}$
These two pieces of data specify the Sierpinski topological space.

Obviously, not everything will be a topological space. For example
1) I give you $X=\{0,1,2\}$
2) I give you $\mathcal{T} = \{\emptyset, \{0,1,2\}, \{0,1\}, \{1,2\}\}$
These are again two pieces of data, but they do not specify a topological space because $\mathcal{T}$ does not satisfy the axioms. Indeed, $\{0,1\}\cap\{1,2\} = \{1\}$ is not in $\mathcal{T}$.

 

[lee.spi]

both “a distinguished collection of subsets”and “to be called the open sets” refer to the set T ?

 

[micromass]

both “a distinguished collection of subsets”and “to be called the open sets” refer to the set T ?

Yes, $\mathcal{T}$ is the "distinguished collection of subsets". Any element of $\mathcal{T}$ is by definition an open set.

 

[lee.spi]

Yes, $\mathcal{T}$ is the "distinguished collection of subsets". Any element of $\mathcal{T}$ is by definition an open set.

now i understand,thanks