Originally Posted by
Porter1 I need help with another complex problem in a general topological space:
Show that a set S is open if and only if each point in S is an interior point.
By writing down two definitions the above statement follows quite easily:
Def 1 x is interior point of S <=> there exists ε>0 and
Def 2 S is open <=> for all ,x: xεS => there exists ε>0 and
.
And using the two definitions we have:
S is open <=> (for all ,x : xεS => there exists ε>0 and
) <=> (for all ,x : xεS => x is interior of S) <=> (each point of S is interior)