Topology

[oxymoron]

This thread follows from what was being discussed in the Metric Spaces thread.

CLOSED SETS
Closed sets are not necessarily open sets.
A set A is closed if X\A is open in X.

Consider the set B = (-∞ , 0) ∪ (1, ∞ ) on the real line. This set is open right? Because for every element in the set, there is a neighbourhood with a finite δ > 0 such that the neighbourhood is still in the set.
But notice that B = R\A where A would be [0,1].

This is one example of a closed finite set. But what is interesting is that the theorem:
In any topological space...
1. The intersection of any collection of closed sets is closed.
2. The union of any finite collection of closed sets is closed.
This is exactly the opposite of the theorem for open sets!!

___

There is a question where if X = {a,b,c,d} and let τ = {Ø , {a}, {a,b}, {a,c}, {a,b,c}, X} be a topology on X.

1) In the topological space (X,τ ) the set {c} is not open because {c} is not in τ. But {c} is not closed either since X\{c} = {a,b,d} is not in τ. Hence I conclude that {c} is neither open nor closed.

2) The set {d} is not open because {d} is not in τ. But X\{d} = {a,b,c} ∈ τ. Hence {d} is not open but is closed.

3) Similarly the set {c,d} is closed in X because X\{c,d} = {a,b} ∈ τ

Am I correct in my working?

[lethe]

This thread follows from what was being discussed in the Metric Spaces thread.

CLOSED SETS
Closed sets are not necessarily open sets.
A set A is closed if X\A is open in X.

Consider the set B = (-∞ , 0) ∪ (1, ∞ ) on the real line. This set is open right? Because for every element in the set, there is a neighbourhood with a finite δ > 0 such that the neighbourhood is still in the set.
exactly. and like I said before, that is often the easiest and most available way to check openness.
But notice that B = R\A where A would be [0,1].

This is one example of a closed finite set. But what is interesting is that the theorem:
In any topological space...
1. The intersection of any collection of closed sets is closed.
2. The union of any finite collection of closed sets is closed.
This is exactly the opposite of the theorem for open sets!!
Right! there is a whole duality between open and closed sets, all provided by de Morgan's laws. All statements about open sets can be translated into statements about closed sets. We could have started with closed sets as well, if we wanted. (note that the axioms of a topology are not a theorem. they are axioms. but the equivalence of the open set axioms to the closed set axioms is a theorem)


There is a question where if X = {a,b,c,d} and let τ = {Ø , {a}, {a,b}, {a,c}, {a,b,c}, X} be a topology on X.

1) In the topological space (X,τ ) the set {c} is not open because {c} is not in τ. But {c} is not closed either since X\{c} = {a,b,d} is not in τ. Hence I conclude that {c} is neither open nor closed.
right, a set can be open and not closed, closed and not open, both open and closed or neither open nor closed.

2) The set {d} is not open because {d} is not in τ. But X\{d} = {a,b,c} ∈ τ. Hence {d} is not open but is closed.
yep.

3) Similarly the set {c,d} is closed in X because X\{c,d} = {a,b} ∈ τ

Am I correct in my working?
you've got it perfectly.