Let [math]X[/math] consist of four elements: [math]X= \{a, b, c, d\}[/math]. Which of the following collections of its subsets are topological structures in [math]X[/math]?

 

[math]1. \emptyset , X, \{a\} , \{b\} , \{a, c\} , \{a, b, c\} , \{a, b\};[/math]

[math]2. \emptyset , X, \{a\} , \{b\} , \{a, b\} , \{b, d\};[/math]

[math]3. \emptyset , X, \{a, c, d\} , \{b, c, d\}?[/math]

 

Are they all topological structures in X? If they are not, why are they not?

Recall the definition of a topological space. A topological space is a set [math]X[/math] together with a collection of subsets [math]\tau[/math] of [math]X[/math] that satisfy the following

 

a) The empty set and [math]X[/math] are in [math]\tau[/math].

b) The union of any collection of subsets in [math]\tau[/math] are also in [math]\tau[/math].

c) The intersection of any (finite) collection of subsets in [math]\tau[/math] are also in [math]\tau[/math].

 

[math]\tau[/math] is referred to as a topology.

 

Now go through your suggested topologies and see if they satisfy the above axioms. Let us know what you find out.