Page 129 - DMTH503_TOPOLOGY

Unit 12: Components and Local Connectedness

12.4 Keywords Notes

Connected: A topological space X is said to be connected if it cannot be written as the union of
two disjoint non-empty open sets.

Discrete Space: Let X be any non empty set of T be the collection of all subsets of X. Then T is
called the discrete topology on the set X. The topological space (X, T) is called a discrete space.
Open Set: Let (X, T) be a topological space. Any set A T is called an open set.

Partition: A topological space X is said to be disconnected if there exists two non-empty separated
sets A and B such that E = A  B. In this case, we say that A and B form a partition of E and we write
E = A/B.
12.5 Review Questions

1. Let p : X Y be a quotient map. Show that if X is locally connected, then Y is locally
connected.
2. A space X is said to be weakly locally connected at x if for every neighbourhood U of x,
there is a connected subspace of X contained in U that contains a neighbourhood of x. Show
that if X is weakly locally connected at each of its points, then X is locally connected.
3. Prove that a space X is locally connected if and only if for every open set U of X and each
component of U is open in X.
4. Prove that the components of X are connected disjoint subspaces of X whose union is X,
such that each non-empty connected subspace of X intersects only one of them.

12.6 Further Readings

Books J.L. Kelly, General Topology, Van Nostrand, Reinhold Co., New York.
S. Willard, General Topology, Addison-Wesley, Mass. 1970.

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