Questions about the definition of open sets

[V0ODO0CH1LD]

I am currently reading Munkres' book on topology, in it he defines an open sets as follows:
"If X is a topological space with topology T, we say that a subset U of X is an open set of X if U belongs to the collection T."

Firstly, are the open sets a property of the set X or the topological space (X,T)? Because if the open sets are all the things in the collection T of a particular topological space, different topological spaces on the same set would have different open sets. Making the whole notion of open sets of a set ill defined.

Which brings me to my next question: are the open sets the sets in all possible collections T for a set X (in which case defining the open sets of a set makes sense), or are the open sets of a topological space the sets in the collection T of that topological space? In the second case I guess saying the open sets of X would have the fact that the open sets are a property of (X,T) implied. Since also the collection T itself is implied.

 

[pasmith]

I am currently reading Munkres book on topology, in it he defines an open sets as follows:
"If X is a topological space with topology T, we say that a subset U of X is an open set of X if U belongs to the collection T."

Firstly, are the open sets a property of the set X or the topological space (X,T)?

Openness is a property of topological spaces. A set $U \subset X$ is open in $(X,\mathcal{T})$ if and only if $U \in \mathcal{T}$.

 

[economicsnerd]

To elaborate:

Given a set $X$, there's no innate meaning to an "open subset of $X$". When somebody refers to, for example, an open subset of $\mathbb R$, what they mean is an element of $\mathcal T_{\mathbb R}$, which is just the usual (i.e. the one defined by absolute value) distance on $\mathbb R$.

 

[Jorriss]

Firstly, are the open sets a property of the set X or the topological space (X,T)? Because if the open sets are all the things in the collection T of a particular topological space, different topological spaces on the same set would have different open sets. Making the whole notion of open sets of a set ill defined.

It's not ill defined, there is just more than one possible topology on a set X.

 

[PSarkar]

Which brings me to my next question: are the open sets the sets in all possible collections T for a set X (in which case defining the open sets of a set makes sense), or are the open sets of a topological space the sets in the collection T of that topological space? In the second case I guess saying the open sets of X would have the fact that the open sets are a property of (X,T) implied. Since also the collection T itself is implied.

Its the second case. Since in general there are many topological spaces for one set, $X$, so the open sets of a topological space of $X$ will depend on which topological space of $X$ you are working with.

 

[1MileCrash]

Open sets are completely determined by the topology on the set. Consider the indiscrete topology; only the empty set and the set itself are open (clopen). Why? Because that collection of the empty set and the point set itself meet all the requirements of being a topology, and we call the members of a topology "open sets."


If you've taken linear algebra - it's exactly how "vectors" are just members of a set we can call a "vector space." In topology, "open sets" are just member of a collection of sets that we can call a "topology."

If you can construct of a family of subsets of X such that it includes X, includes the empty set, and includes any union and finite intersection of other members of the family, you have a topology, and its members are called open sets.