Dictionary order topology

Show that the dictionary order topology on the set RxR....

Criticism of a proof

Criticize the following proof...(see attachment)

Hausdorff

Show that X is Hausdorff if and only if... --- (See attached file for full problem description)

Neighborhoods

Show that the T1 axiom is equivalent to... --- (See attached file for full problem description)

General and Differential Topology

Content: Topological and metric spaces, continuity, subspaces, products and quotient topology, compactness and connectedness, extension theorems, topological groups, topological and differentiable manifolds, tangent spaces, vector fields, submanifolds, inverse function theorem, immersions, submersions, partitions of unity, Sard' ...continues

Topology proof

Suppose that f:X->Y is continuous.... --- (See attached file for full problem description)

Continuous function proof

Let f:rxr-->r be defined by the equation.... --- (See attached file for full problem description)

Uniquely determined proof

Let f:A--Y be continuous with Y Hausdorff.... --- (See attached file for full problem description)

General and Differential Topology

Find conditions under which the box topology is strictly finer than the product topology. See attached file for full problem description.

Topology proofs

Let A be a set. Let {Xa} be a family of spaces.... --- (See attached file for full problem description)