# topology

1. ### Arbitrary intersection of open sets

Arbitrary intersection of the open set (-1/n,1/n) where n is a natural number is zero. Also the arbitrary intersection of the set (-r,r) where r is a positive real numbers is zero. Could someone please explain, why is it like this ?
2. ### Basic topology question.

Assume X is a metric space, and X = A\cup B where A and are closed subspaces. Can a boundary point of A or B be a boundary point of A\cup B ? My guess is that it can be, that a metric space can have a boundary. But if that's so, I am misunderstanding something else: the textbook problems I...
3. ### Isometry composition, (f^n (x)) - increasing sequence

Let f: [0,1] \rightarrow [0,1] be an isometry in a metric space ([0,1], d_{Eucl}). Suppose f(0)=0, \ \ \ x\in [0,1] Could you explain to me why the sequence (f^n(x))_{n\geq1} is monotone and why does it converge with respect to the usual topology of [0,1] and also with respect to d and, in...
4. ### question about topology in R^2

hello, i'm looking for example for a closed group in R^2, but its projection on R is not closed. any ideas?
5. ### Topology Winding Number Vector Field Question

Could someone please clarify an answer for me to the following question: Calculate the winding number of the vectorfield: [x^3 + 3x(y^2) , 3x(y^2) - (y^3)] along the curve x^2 + y^2 = 1 I have tried sketching it out for the the +-1 coordinates of x and y and for the diagonals where (x,y) =...
6. ### Topology

Prove that [0,1] and (0,1) are not homeomorphic by showing that any continuous map f: [0,1] --> (0,1) is not onto. Similarly, prove that [0,1), [0,1], (0,1) are not homeomorphic to each other. I have no idea how to do it. Can anyone tell me?
7. ### Metric Space and Topology HW Help!!!

Let X be a metric space and let (sn )n be a sequence whose terms are in X. We say that (sn )n converges to s ∋ X if ∀ ϵ > 0 ∃ N ∀ n ≥ N : d(sn,s) < ϵ For n ≥ 1, let jn = 2[(5^(n) - 5^(n-1))/4]. (Convince yourself that 5^(n) - 5^(n-1) is always divisible by 4, so the exponent in the definition...
8. ### 2 topology problems

Hi. Can anybody help me with this problems: 1. Prove that S^1 \ N is homeomorphic with R. S^1 = {(x,y)|x^2 + y^2=1}, N=(0,1) (S^1\N is circular without one point, (0,1)) 2. If set A is countable subset of R^2 (of plane), prove that R^2 \ A is path connected.
9. ### Topology question

Hey sorry for another post. I'm trying to do an assignment and I really don't understand this question A intersection B is compact. Let B a subset of R2, be as follows: B = { (x,y) in R2s.t. y = sin(1/x), x>0} U {0,0}. a. Is B closed?, open?, bounded?, compact? Can you please...
10. ### Initial topology

Hi, In my course topology there's a propisition which says the following: suppose f and g are continuous functions and f \circ g is initial then g is initial (I mean with initial that f \circ g and g have the initial topology on their domains). Now, they ask for a counterexample. I have to...
11. ### looking for topology and category theory

Hi, I am looking to understand and learn topology. I am also want to learn category theory. I know what category theory is basically but want to understand it better and so that interpretations can be derived from its theorem.
12. ### Compact in R

Hello! I was wondering if someone could clear something up for me.. Is every non-empty, compact subset of R a closed interval in R? It seems to me that this is true. I know that such sets are closed an bounded, but I don't know how to show that they're connected. Any help is much appreciated!
13. ### Topology. Matrix question

Let x_1 , x_2, ..., x_k be vectors in \mathbb{R}^n; let X be a matrix X = [x_1 , x_2, ..., x_k]. If I = (i_1, i_2,...,i_k) is an arbitrary k-tuple from the set \{1, 2, ... ,n\}, show that \phi_{i_1} \wedge \phi_{i_2} \wedge ... \wedge \phi_{i_k} (x_1 , x_2, ..., x_k) = det X_I, where...
14. ### Topology. Computational exercise.

Let U = \mathbb{R}^2 \setminus \{(0,0)\}; consider the 1-form in U defined by \omega = \dfrac{x dx + ydy}{x^2+y^2} (a) Show that d\omega =0 (b) Show that there is a 0-form \theta on U, such that d\theta = \omega Please help!!!
15. ### Topology. Elementary alternating tensors.

Let x,y,z,w \in \mathbb{R}^5 Let T(x,y,z) = 2x_2 y_2 z_1 +x_1 y_5 z_4 S(x,y) = x_1 y_3 + x_3 y_1 R(w) = w_1-2w_3 (a) Write Alt(T) and Alt(S) in terms of elementary alternating tensors. (I think this is analogically to writing T and S in terms of alternating tensors ) (b) Express...
16. ### Topology: How to formally write proofs?

Hey there! In doing practice exercises for topology, I find that I can reason through them easily enough to find the correct answer. The problem is that I rely predominantly upon verbal logic in order to make the proofs work. How does one go about proofs using only the math notation? A simple...
17. ### Is it a base for a Topology in R^N

Hi there Prove this statement or the opposite: Consider the set \mathbb{R}^{\mathbb{N}} := \{ (x_n)_{n \in \mathbb{N}} : x_n \in \mathbb{R} \forall n \in \mathbb{N} \} . Define U_{y_1,...,y_m}(r):= \{ (x_n)_n \in \mathbb{R}^{\mathbb{N}} : |x_i - y_i| < r \text{ for every } i=1,...,m \} . The...
18. ### Topology Proof

If T1 and T2 are topologies for X and B={(G1∩G2)|G1∈T1, G2∈T2}, Then T*={G|For all p, if p∈G, then A∈B such that p∈A⊆G} is a topology. So I need to show (1) T*⊆P(X) (2) ∅∈T* and X∈T* (3) if A,B∈T*, then A∩B∈T* (4) if A⊆T*, then ∪A∈T* This is part of a problem proving T~=∩{T|T is a topology...
19. ### Topology Prove continuous using bases

Need some help proving Claim2.1 g1:X1xX2→ X1 and g2:X1xX2→ X2 Claim 2: if (g1 o f): E → X1 and (g2 o f): E → X2 are both continuous, then f: E → (X1 x X2) is continuous. Proof: Let (g1 o f): E → X1 and (g2 o f): E → X2 both be continuous We know {(G1xG2)| G1∈Τ1 and G2∈Τ2 } is a base for ΤX1 x X2...
20. ### f: E → (X1 x X2) is cont iff (g1 o f): E → X1 and (g2 o f): E → X2 are both cont

I have an outline for the proof, but would appreciate someone checking if the logic in Claim 1.1 is sound and I could use some help with Claim 2.1 For any topological space (E, ΤE) and any function f: E → (X1 x X2), f: E → (X1 x X2) is continuous iff (g1 o f): E → X1 and (g2 o f): E → X2 are...