# homeomorphisms

1. ### Homeomorphisms and Topologies

Wanted to make sure I have all the technical details in the following argument correct Let (X_1,T_1) and (X_2,T_2) be topological spaces and f:X_1\rightarrow{X_2} be a bijection. Prove that f is a homeomorphism if and only if f(T_1)=T_2 Proof Let f be a homeomorphism. Consider U\subset{X_1}...
2. ### Homeomorphisms

Let f: (M_1,d_1)\to(M_2,d_2) and G(f)=\{(x,f(x))|x\in M_1\}\subseteq M_1\times M_2 and let \tilde f: (M_1,d_1)\to(G(f),d_p). Consider \tilde f(x)=(x,f(x)) and d_p((x_1,x_2),(y_1,y_2))=\sqrt{d_1(x_1,y_1)^2+d_2(x_2,y_2)^2} for all (x_1,y_1),(x_2,y_2)\in M_1\times M_2. Prove that \tilde f is a...
3. ### compactness of sets of homeomorphisms of a group.

Hello; Let M be a manifold and let U be a nonempty euclidean open subset of M. Let B be a nonempty open ball whose compact closure cl(B) is contained in U. Let H(cl(B)) be the group of homeomorphisms of cl(B). Then prove that the set of all functions in H(cl(B)) that fix the boundary ∂B of B...
4. ### Topology question: continuous maps and homeomorphisms

Let R denote the real numbers. Let S^2 denote the unit sphere in R^3 Let f: S^2 -> R^4 be defined by f(x,y,z)=(x^2-y^2,xy,yz,zx) Can we prove that: f determines a continuous map g: PR^2 -> R^4 where PR^2 is the real projective plane. & g is a homeomorphism onto a topological subspace of R^4
5. ### Topology homeomorphisms and quotient map question

In this R denotes the real numbers, and [.] denotes subscript Let Homeo(X) be the group of homeomorphisms f:X->X where X is a topological space. Let G be a subgroup of Homeo(X) ~ is an equivalence relation on G where x~y iff there exists g in G s.t g(x)=y Q1: Let p: X ->...
6. ### proving homeomorphisms, circles, torus, square

hey, i have trouble proving homeomorphisms, i am new to this concept. the problem is: The circle {(0, x2, x3): (x2-1)^2 + (x3)^2 = 1} lies in the plane x1=0 in R^3. From the quotient space from the square [0,1]x[0,1] contained in R^2, show that this quotient space is homeomorphic to the...
7. ### Homeomorphisms..

Problem: Construct a homeomorphism \phi : \{0,1\}^{\mathbb{N}} \to \{0,1,2\}^{\mathbb{N}} with the product topology on both spaces. Tried thinking of a few simple examples so far, but none were satisfactory. Any clues\hints? Thanks, and happy holidays.
8. ### Quick question... homeomorphisms...

Can a homeomorphism exist between an open and a half open set? (ie: (0,1) and [0,1)) I know that to be a homeomorphism, a bijection must exist, they must be continuous, and they must have a continuous inverse... Where to go from here? The fact that 0 is not included within the first set...
9. ### Continuous functions and homeomorphisms

Let X and Y be topological spaces and f : X \longrightarrow Y a function. The graph of f is the set G_f = \{(x,f(x)) : x \in X\}. Prove that f is continuous if and only if the function \Phi : X \longrightarrow G_f given by \Phi(x) = (x,f(x)) is a homeomorphism where G_f has the topology...
10. ### Homeomorphisms and cutsets

Let f:X \rightarrow Y be a homeomorphism. Prove: If S is a cutset of X, then f(S) is a cutset of Y
11. ### homeomorphisms and interior, boundary

Show that if f: X->Y is a homeomorphism, then: f(\partial(A))=\partial(f(A)) I am stuck!