Reference: Hocking and Young’s Topology, Dover Publishers. Chapter 1: Topological Spaces and Functions.
Definition : Separated Space: A topological space is separated if it is the union of two disjoint, non empty, open sets.
Definition: Connected Space: A topological space is connected if it is not separated.
PS: Both separatedness and connectedness are invariant under homeomorphisms.
Lemma 1: A space is separated if and only if it is the union of two disjoint, non empty closed sets.
Lemma 2: A space S is connected if and only if the only sets in S which are both open and closed are S and the empty set.
Theorem 1: The real line is connected.
Theorem 2: A subset X of a space S is connected if and only if there do not exist two non empty subsets A and B of X such that , and such that is empty.
Note the above is Prof. Rudin’s definition of connectedness.
Theorem 3: Suppose that C is a connected subset of a space S and that os a collection of connected subsets of S, each of which intersects C. Then, is connected.
Corollary of above: For each n, is connected.
Every continuous image of a connected space is connected.
Lemma 3: For , the complement of the origin in is connected.
Theorem 5: For each , is connected.
Theorem 6: If both and are continuous, then the composition gf is also continuous.
Lemma 4: A subset X of a space S is compact if and only if every covering of X by open sets in S contains a finite covering of X.
Theorem 7: A closed interval in is compact.
Theorem 8: Compactness is equivalent to the finite intersection property.
Theorem 9: A compact space is countably compact.
Theorem 10: Compactness and countable compactness are both invariant under continuous transformations.
Theorem 11: A closed subset of a compact space is compact.