Abstract_Logic Posted January 4, 2010 Share Posted January 4, 2010 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? Link to comment Share on other sites More sharing options...
ajb Posted January 4, 2010 Share Posted January 4, 2010 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. Link to comment Share on other sites More sharing options...
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