Do arbitrary topological spaces have competing notions of topological dimension?

For arbitrary topological spaces there are competing notions of topological dimension. In “nice
spaces, the di?erent notions of topological dimension tend to coincide. However, even the well- behaved euclidean spaces, Rn , n > 1, have surprises in store for us.

What are the best books on algebraic topology?

Molecular topology. Nova Science Publishers, Huntington, N.Y., 2001. C.T.J. Dodson. Category bundles and spacetime topology. Kluwer Academic Publishers, Dordrecht, Boston, 1988. C.T.J. Dodson. A user’s guide to algebraic topology. Kluwer Academic Publishers, Dordrecht, Boston, 1997. Albrecht Dold. Lectures on algebraic topology.

What are topological notions?

Topological notions like compactness, connectedness and denseness are as basic to mathematicians of today as sets and functions were to those of last century.

How many topologies can be put on a set?

In Chapter 1 we de?ned three topologies that can be put on any set: the discrete topology, the indiscrete topology and the ?nite-closed topology. So we know three topologies that can be put on the set R. Six other topologies on R were de?ned in Exercises 1.1 #5 and #9.