Question about how a topology is defined I have an exercise that defines a topology this way:
$$\tau=\{\emptyset\}\cup\{\{1,2,3,...,n\}:n\in\mathbb{N}\}\cup\{\mathbb{N}\}$$
I understand that in $\{\{1,2,3,...,n\}:n\in\mathbb{N}\}$ there's only one set, and it depends on what $n$ we choose. For example, if we take $n=3$, our topology is $$\tau=\{\emptyset\}\cup\{1,2,3\}\cup\{\mathbb{N}\}.$$
Am I right? If not, how this has to be understood?
Thanks!
 A: 
I understand that in {{1,2,3,...,n}:n∈N} there's only one set, and it depends on what n we choose. 

No.  You understand wrong and that is exactly what it doesn't mean.
That would be $\{\{1,2,3,..... n\}:$ for some  $n \in \mathbb N\}$.
It is understood that this notation means $\{\{1,2,3,.... n\}:$ for every $n \in \mathbb N\}$
So this set is infinite  and is equal to $\{\{1\}, \{1,2\}, \{1,2,3\}...... \{1, 2, 3...... k\}, \{1,2,3,......., k+1\},.......\}$.
A: The variable $n$ is just an index variable.

So the topology has open sets of the form $\{1,2,3,...,n\}$ for each positive integer $n$.

Thus, the topology is
$$\tau=\{{\large{\varnothing}},\mathbb{N},\{1\},\{1,2\},\{1,2,3\},...\}$$
Of course, you should check to make sure the axioms are satisfied.


*

*Is the empty set open?$\\[4pt]$

*Is the whole space $\mathbb{N}$ open?$\\[4pt]$

*Is the intersection of two open sets open?$\\[4pt]$

*Is an arbitrary union of open sets open?$\\[4pt]$


The checks are routine, but you should at least think them through.
