topology

1. Intro to topology and open sets

Hello All, Topology is just getting under way for me, and I have a question. Consider a set {n}. Is it open, closed, or neither? Specifically, say {8}, for example. I say this set is closed, because it has upper and lower bounds at {8}. Also, as I understand it, the compliment {(-inf,8) u...
2. Show that set is not open nor closed

I have a set I = {x from R2 : x1<1 v x2>=2} I have to show it isn't neither open or closed. Can I do this?: x1<1then I can choose for that point an open disc of radius r = 1-x1 Every point y in that disc has 2x1−1<y1<1, so y1<1, so every point in the disk satisfies the first inequality so...
3. Limit definition in Rn

I have a question. Let f : Rn → Rm and let l ∈ Rm. Show that lim x→c f(x) = l ⇔ lim x→c||f(x)−l|| = 0. How can I do this? I know the (ε, δ)-definition of limit, but this is in Rn. Thank you
4. Basis of a Topology

I am presently reading about the concept of a topological basis, and I seem to be encountering conflicts between various sources. The definition of basis I am working with is the following: If X is a set, a basis for a topology on X is a collection \mathcal{B} of subsets of X such that (1)...
5. Confusion in Topology definition

In definition of topology, why we say intersection of finite members of τ belongs to τ. What is harm in any number of members of τ in it?
6. Is orientable surface with a full twisting homeomorphic to one with no full twisting

Hi Suppose X is an orientable closed surface with some full twisting. Is there a surface D such that D is homeomorphic to X and D has no full twisting? I think the answer is yes. This is because any aorientable closed surface can be desrcibed as a sphere with some 1-handles attached to...
7. Topology

Let X=[0,1]x[0,1]. We endow the set X with the order topology using the dictionary order in [0,1]x[0,1]. Find the closure on X of A={1/n x 0 : n in Z} I know that the closure is equal to the set A union the point (0,1), but can´t find an easy way to sort of prove that any other point won't...
8. Metric and topology. The balls.

I will use the lemma: i) If $b\in K(a,r)$ and $0<s\leq r-d(a,b)$, then $K(b,s)\subseteq K(a,r)$. ii) If $K(a,r)\cap K(b,s)\neq \emptyset$, then $d(a,b)<r+s$. a) Assume that there exist $m\in K(c,t)$. I want to show that $d(a,m)<r$. Since $c\in K(b,s)$, so $d(a,c)\leq d(a,b)+d(b,c)<d(a,b)+s$...
9. Topology equivalence in dynamical system

Hi, my name is Eric. I've got trouble when proofing that system \dot{x}=\alpha+x^2+O(x^3) is topological equivalence with system \dot{x}=\alpha+x^2. I don't understand how to build the homeomorphism for the orbit. In literature, I read that for \alpha>0, the homeomorphism mapping is defined by...
10. Topology

a, b belong R3 d(a,b) = ((a1 - b1)2 + (a2 - b2)2 + (b3 - b3)2)1/2 (R3, d) is a metric space. For x belongs R3, r > 0, Ur(x) = { y belongs R3 : d(y,x) < r}, the open ball of radius r centered at x. let T be the set of all subsets U of R3 if x belongs U, then there is r > 0 such that Ur(x)...
11. topology of wheel with n spokes

Hi, The wheel with n spokes is not homeomorphic to the wheel with m spokes (in the subspace topology in the plane).Why is that? Thank's in advance

Hi, I am working through Bert Mendelson's "Introduction to Topology" and am having some trouble with proofs. The text in well presented but to get a proper understanding I am working through the excercises. I am still on Chapter 1 which deals with sets and for the most part the excercises seem...

14. Mobius Band as a Quotient Topology

I am reading Martin Crossley's book, Essential Topology. I am at present studying Example 5.55 regarding the Mobius Band as a quotient topology. Example 5.55 Is related to Examples 5.53 and 5.54. So I now present these Examples as follows: I cannot follow the relation (x,y) \sim...
15. Simple topology problem involving continuity

Example 1 in James Munkres' book, Topology (2nd Edition) reads as follows: Munkres states that the map p is 'readily seen' to be surjective, continuous and closed. My problem is with showing (rigorously) that it is indeed true that the map p is continuous and closed. Regarding the...
16. continuity of a function

Hello; I have a question about continuous functions; Let A={2,4,6} and B={1,3,5}. A and B are two discrete subsets of the real line so that the topology of them is the induced topology of the real line. Define a function f from A to B such that f(2)=1, f(4)=3 and f(6)=5. Is the function f...
17. initial topology with respect to norm

Hello, Suppose (X,\|.\|) is a normed vector space. Let (X,\mathcal{T}) be the initial topology with respect to \|.\|: X \rightarrow [0,\infty). Then the topology induced by the norm is not equal to \mathcal{T}. I've been thinking about this for some time now but I'm having real trouble proving...
18. Final segment topology and finite-closed topology?

I'm trying to answer a question I found in an elementary topology textbook. I keep feeling I'm missing something. Suppose N is the set of all positive integers and t consists of N, the empty set, and every set { n, n+1, ... } for n any positive integer. This is a topology and is called the...
19. Finite Closed Topology

As I understand it, a topology T on a set X (finite or infinite) is called "the finite-closed topology" or "the cofinite topology" if the closed subsets of X are X itself, plus all the finite subsets of X; that is, the open sets are the empty set and all subsets of X which have finite...
20. Topology proof

Prove that S^(int), boundary of S and (S^c)^int are pairwise disjoint and their union is the entire Rn. I understand that the set S is separate from it's boundary line then (S^c)^int is everything excluding the other two but how do I go about the proof?