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.