
Trivial Topology, Continuity and Connectedness
57806 Trivial Topology, Continuity and Connectedness Let X and Y be topological spaces, where the only open sets of Y are the empty set and Y itself, i.e., Y has the trivial topology.
? Show that any map X > Y is continuous
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Covering Spaces : Compact Hausdorff Spaces and Homomorphisms
(proof1)
Let and Y be topological spaces and suppose there is a surjective continuous map f : X Y which satisfies the following condition: for each x Y, there is an open neighborhood of such that
• f 1 (U) is a disjoint union of open sets A

Topological Spaces and Continuity
Topological spaces and continuity are investigated. The solution is detailed and well presented.

Countable collections
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

Path components
This solution is comprised of a detailed explanation to show that for any continuous map between topological spaces, there exists a map such that the following holds:
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? for two continuous maps and we have
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Path connected problems
Show that if f:X>Y is a continuous map between topological spaces and X is path connected, then the image f(Y) is also path connected.
 Please see the attachment. This solution is comprised of a detailed explanation to
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Linear Mapping in Subsets
(X).
7) If alpha: X_1 > X_2 and beta: X_2 > X_3 are continuous functions of compact topological spaces, explain why (beta is in alpha)* = alpha* is in beta*
8) Hence prove that if gama: X > Z is a homeomorphism of compact topological spaces

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

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