Topological spaces

Concepts

• Topology 10 exercisesA topology on a set X is a family T of subsets of X, whose members U are called open sets, containing ∅ and X and closed under ar…
• Topological space 10 exercisesA topological space is a set X together with a topology T on it; the members of T are its open sets U.
• Discrete and trivial topology 10 exercisesThe discrete topology T_disc on a set X has the power set P(X) as open sets; the trivial one T_triv, only the empty set and X.
• Closed set 10 exercisesA subset C of a topological space (X,T) is closed when its complement in X is an open set of the topology T.
• Basis of a topology 10 exercisesA basis B of a topology T on X is a family of open sets such that every open set U of T is a union of members of B.
• Neighborhood of a point 10 exercisesA neighborhood of a point x in a topological space (X,T) is a set N⊆ X that contains some open set UinT to which x belongs.
• Interior of a set 10 exercisesThe interior intA of a subset A of a topological space (X,T) is the largest open set UinT contained in A.
• Closure of a set 10 exercisesThe closure overlineA of a subset A of a topological space (X,T) is the smallest closed set F of X containing A.
• Boundary of a set 10 exercisesThe boundary ∂ A of a subset A of a topological space X is the set of points of the closure of A that do not belong to its interi…
• Accumulation point 10 exercisesA point x of a topological space X is an accumulation point of a subset A if every neighborhood N of x contains a point of A othe…