Some basic facts about connectedness and compactness

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 E^{1} 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 X = A \bigcup B, and such that (\overline{A} \bigcap B) \bigcup (A \bigcap \overline{B}) 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 \{ C_{\alpha}\} os a collection of connected subsets of S, each of which intersects C. Then, S = C \bigcup (\bigcup_{\alpha}C_{\alpha}) is connected.

Corollary of above: For each n, E^{n} is connected.

Theorem 4:

Every continuous image of a connected space is connected.

Lemma 3: For n>1, the complement of the origin in E^{n} is connected.

Theorem 5: For each n>0, S^{n} is connected.

Theorem 6: If both f: S \rightarrow T and g: T \rightarrow X 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 [a,b] in E^{1} 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.

Cheers,

Nalin Pithwa.

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