topology

  1. K

    Topology Proof: g:(X1 x {p2})→ X1 is a homeomorphism

    I've outlined the proof but could use some help with the details in the simpler sections (eg: proving a bijection) For any p2 ∈ X2 π1|(x1x{p2}): (X1 x {p2})→ X1 is a homeomorphism (ie: bijective, continuous and open) Proof: Consider any p2 ∈ X2 Let g=π1|(x1x{p2}) Claim 1: g: (X1 x {p2})→ X1...
  2. V

    Topology. Metric Space.

    Picture attached.
  3. V

    Topology exercise.

    Picture attached.
  4. K

    Base Topology Proof

    For topologoical spaces (Τ1,X1) and (Τ2,X2), {(G1 x G2)| G1 ∈ Τ1; G2 ∈ Τ2 is a base for the product topology T(X1 x X2) We know: B is a base for Τ iff (1) B⊆Τ and, (2) For every G∈Τ, if p∈G, then there is B∈B such that p∈B⊆G.
  5. K

    Topology proof: smallest topology and base for topology

    Thm: If Τ1 and Τ2 are topologies for X and Τ*=∩{ Τ | Τ is a topology for X such that Τ1 ⊆ Τ, Τ2 ⊆ Τ}, then (1) Τ* is the smallest topology for X containing both Τ1 and Τ2; and (2) {(G1∩G2)| G1 ∈ Τ1, G2 ∈ Τ2} is a base for Τ* We know: B is a base for Τ iff (1) B⊆Τ and, (2) For every...
  6. G

    topology, metric spaces, proving continuity

    After seeing all the other topology problems be left out to dry, I'm not exactly confident Ill get help, but ill try anyway. So heres a photo of my problem, since im not the best at latex ok i get that we have to start with let \epsilon > 0 then we should have \delta > 0 as well right? well...
  7. C

    topology ans sequences

    Can you find any sequence of rationals such that it converges to π/4?use the fact that the rationals are dense in the reals, or you can think about the decimal expansion of π/4. How can you get rational numbers out of that, rational numbers that converge to π/4? Can anybody help me with this...
  8. C

    topology

    Let X be a normed space.Show that the function f:X to Real numbers defined by f(x)=norm x is continuous on X.
  9. C

    Topology problems

    can somebody help me with these two problems immediately??? 1.Let D_p be any open disc in R×R with centre p=(a,b) and radius δ>0.Show that there exist an open disc (Dbar) such that (i)the centre of (Dbar) has rational coordinates. (ii) the radius of (Dbar0 is rational. (iii)p is an element of...
  10. V

    Problem on product topology/standard topology on R^2.

    Let $\mathbb{R}_{\tau}$ be the set of real numbers with topology $\tau = \{(-x,x)| x>0\} \cup \{\emptyset, \mathbb{R}\}$ and $\mathbb{R}_{\tau} \times \mathbb{R}_{\tau}$ be the product topology on $\mathbb{R}^2$. a)Prove that $A = \{(x,y) \in \mathbb{R}^2 | x^2 + y^2 < 1\}$ is open in...
  11. V

    Continuity of fog and f/g on standard topology.

    Let $f: X \rightarrow \mathbb{R}$ be continuous functions, where ($X, \tau$) is a topological space and $\mathbb{R}$ is given the standard topology. a)Show that the function $f \cdot g : X \rightarrow \mathbb{R}$,defined by\\ $(f \cdot g)(x) = f(x)g(x)$\\ is continuous.\\ b)Let $h: X \setminus...
  12. V

    Please help! Homeomorphism in R^2. Standard Topology.

    Let $A = \{(x,y) \in \mathbb{R}^2 | (x+1)^2 + y^2 = 1\}$,\\ $B = \{(x,y) \in \mathbb{R}^2 | (x-1)^2 + y^2 = 1\}$, and\\ $C = \{(x,y) \in \mathbb{R}^2 | x^2 + y^2 = 1\}$.\\ Define $N = A \cup B$ and let $N$ and $C$ be given subspaces of ${\mathbb{R}}^2$, where ${\mathbb{R}}^2$ is given the...
  13. M

    Topology Boundary of a set is closed? A set subset of it's interior implies open set?

    A is a nonempty set. Show boundary of A is closed. Obviously dealing in the real number space. I've seen a couple of proofs for this, however they involve 'neighborhoods' and/or metric spaces and we haven't covered those. Here are some following terms we've define: segment, limit point...
  14. B

    Topology definition (open/closed)

    Is the set from -infinity to +infinity open, closed, neither and/or both?
  15. K

    Topology Proof

    Prove: For any topologies T1 and T2 for X, if B1 is a base for T1, and if B1 ⊂ T2, then T1 ⊂ T2. Using: Thm 2.1: A is open in (X,T) iff For every p ∈ A, there is G ∈T such that p ∈ G and G ⊂ A Cor 8.2: B is a base for T iff: B ⊂ T and for every G ∈ T, if p ∈G, then there is A ∈ B such that p is...
  16. K

    Topology Proofs, Bases

    1) B={(a,b)|a,b are rational numbers} is a countable base for the usual topology for the real numbers. 2) For any topologies T1 and T2 for X, if B1 is a base for T1 and B1 is a subset of T1, then T1 is a subset of T2 Def1: Bp is a local base for p in (X,T) iff p is an element of B and B is an...
  17. K

    Topology Proofs, Bases

    1) B={(a,b)|a,b are rational numbers} is a countable base for the usual topology for the real numbers. 2) For any topologies T1 and T2 for X, if B1 is a base for T1 and B1 is a subset of T1, then T1 is a subset of T2
  18. K

    Closure in the usual topology

    p is an element of the real numbers. e>0. Prove that [p-e, p+e] is the smallest closed set containing (p-e, p+e) in the usual topology (aka: the closure of (p-e, p+e) = [p-e, p+e]) My attempt: Proof: Consider p an element os the real numbers and e>0. Claim 1: The closure of (p-e, p+e) in the...
  19. K

    Topology Closed Neighborhood Proof

    (8) N is a closed neighborhood of p in R_U iff N=R Where R represents the real numbers and R_U is the upper topological space for the real numbers (R, T_U) T_U = {A is a subset of R | a is an element of A implies (a - e, +inf) is a subset of A for some e > 0} (upper topology) I proved "=>"...
  20. K

    Prove Closed Neighborhoods in R's Topologies

    (1) For any point p, where p is an element of R, {p} is an open-and-closed neighborhood of p in R* R represents the real numbers R*={A is a subset of R|a is an element of A implies {a} is a subset of A} (discrete topology) (2) For any point p, where p is an element of R, N is a closed...