Page 121 - DMTH503_TOPOLOGY

Unit 11: Connected Spaces, Connected Subspaces of Real Line

11.5 Review Questions Notes

1. Let {A } be a sequence of connected subspaces of X, such that A A   for all n. Show
n n n+1
that UA is connected.
n
–1
2. Let p : X  Y be a quotient map. Show that if each set p ({y}) is connected and if Y is
connected, then X is connected.
3. Let Y  X ; let X and Y be connected. Show that if A and B form a separation of X – Y, then
Y A and Y B are connected.
4. Let (X, T) be a topological space and let E be a connected subset of (X, T). If E has a
separation X = A|B, then either E  A or E  B.
5. Prove that if a connected space has a non-constant continuous real map defined on it, then
it is uncountably infinite.
6. Show that a set is connected iff A is not the union of two separated sets.
7. Let f : S’  R be a continuous map. Show there exists a point x of S’ such that f(x) = f(–x).
8. Prove that connectedness is a topological property.
n
n
9. Prove that the space R and C are connected.
11.6 Further Readings

Books William W. Fairchild, Cassius Ionescu Tulcea, Topology, W.B. Saunders Company.
B. Mendelson, Introduction to Topology, Dover Publication.

Online links www.mathsforum.org

www.history.mcs.st/andrews.ac.uk/HistTopics/topology/in/mathematics.htm

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