Connected point sets proof

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• Question about Connected point sets

  This is a proof regarding connected point sets.

• Point set

  3183 Proof Prove true or false, if false give a counter example. If each point in the point set M is a limit point of M, then M is connected. Please see attachment This is a proof regarding limit points and connected sets.

• Topology : Connected Spaces and Explosion Point

  A topological space is connected if it can not be expressed as the union of two nonempty disjoint open sets. b. Proof: Since is open in and is a subspace in , then we can find an open set in , such that .

• Covering Spaces : Compact Hausdorff Spaces and Homomorphisms

  Proof:1 Consider any point x not in s. For each y in s, embed y in an open set disjoint from an open set containing x. Do this for each y in s, and we have an open cover.

• Connected Spaces : If X is a connected space containing more than one point, and if {x} is closed subset for every x is a member of X show that the number of points in X is infinite.

  54468 Connected space containing more than one point If X is a connected space containing more than one point, and if {x} is closed subset for every x is a member of X show that the number of points in X is infinite.

• Connected Convex Subsets

  (c) Prove that every convex subset of Rk is connected Proof: We should clarify the definition of separated sets. We say and are separated, if the distance of and is .

• Open Sets, Connectivity and Continuous Functions

  The first one I think I got, connected(M) --> connected (R) open sets in f-1 implies open sets in f, and connected definition implies there exists a m. The second is a bit more tricky, I assume its an inverse open set argument.

• Fundamental Groups, Path-Connected Space, Connectivity and Homotopy

  Proof: We are already given that X is a path-wise connected, so it enough to prove the uniqueness of connecting paths. Suppose 1(X,x0) = 0[where x0 is the base point].

• Real Analysis : Connected and Disconnected Sets

  Connected and Disconnected Sets and Closure are investigated. The solution is detailed.