Previous |  Up |  Next


connected; arcwise connected; point arcwise connected; locally connected; cut points; product spaces; long line; arcs; L-arcs; their unions; continuous images and inverse images
As per the title, the nature of sets that can be removed from a product of more than one connected, arcwise connected, or point arcwise connected spaces while preserving the appropriate kind of connectedness is studied. This can depend on the cardinality of the set being removed or sometimes just on the cardinality of what is removed from one or two factor spaces. Sometimes it can depend on topological properties of the set being removed or its trace on various factor spaces. Some of the results are complicated to prove while being easy to state. Sometimes proofs for different kinds of connectedness are similar, but different enough to require separate proofs. Many examples are given to show that part of the hypotheses of theorems cannot be dropped, and some examples describe results about spaces whose connectedness can be established directly but not with the help of our results. A large number of examples are given for such purposes.
[E89] Engelking R.: General Topology. Heldemann Verlag, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[HY61] Hocking J., Young G.: Topology. Addison-Wesley Publishing Co., New York, 1961. MR 0125557 | Zbl 0718.55001
[K68] Kuratowski K.: Topology, volume II. Academic Press and Polish Scientific Publishers, Warsaw, 1968. MR 0259835
[L74] Lehman B.: Products of arcwise connected spaces. Proc. Amer. Math. Soc. 44 (1974), 221-224. MR 0346762 | Zbl 0285.54006
[M75] Munkres J.: Topology, A First Course. Prentice-Hall, Englewood Cliffs, New Jersey, 1975. MR 0464128 | Zbl 0306.54001
[N92] Nadler S.: Continuum Theory. Marcel Dekker Publ. Co., New York, 1992. MR 1192552 | Zbl 0819.54015
[W70] Willard S.: General Topology. Addison-Wesley Publishing Co., Reading, Mass., 1970. MR 0264581 | Zbl 1052.54001
Partner of
EuDML logo