topological

1. Topological space and continuous function

Let $X$ be a toological space. Suppose that $f$ and $g$ are continuous maps $X\to \mathbb{R}$. Define the function $(f+g)(x)$ by $(f+g)(x)=f(x) + g(x)$. Show that $$(f+g)^{-1}((a,b))=\bigcup_{s,t\in \mathbb{R}}f^{-1}((s,t))\cap g^{-1}((a-s,b-t))$$ and deduce that $f+g$ is continuous. Here's...
2. connected sets in the topological space of real n-matrices Mn(R)

Hi, I need help with proving the following two properties: 1)The set of matrices in Mn(R) with det(A)=1 is connected. 2)The set of matrices in Mn(R) with positive determinant,and the set of matrices with negative determinant are connected,and are the two connected components of the invertible...
3. Topological space

Are power sets always a discrete topology? One video lecture I watched introduced a topology as 2^X for some set X. (I know this forum is more for discussion not help but this is extracurricular work so I figured no need to take up space in the forum where people have questions for...
4. Topological space: Condition 3

\for A,B \in \mathfrak{T}, A \cap B \in\mathfrak{T} What is a good way to prove this true? One video I saw said it can be proved by induction but I haven't done an induction proof on a set before. I thought maybe an element proof would be easier to work out.
5. finding a specific topological group with specific conditions

Hi; I have a question, it sounds difficult. The question is the following: Let X be a topological group such that the binary operation defined on it is *. For any two points a and b in X define a new operation by a(*)b=b^-1*a*b, [(*) is a new operation on X inherited from *]. By this (*), we...
6. A Property of Boundary of a Set in a Topological Space

Dear Colleagues, I want a help for proving the following property: Bd[Bd{Bd(A)}]=Bd[Bd(A)], where Bd denotes the boundary and A is subset of a toplogical space X. Actually I proved that Bd[Bd{Bd(A)}] is a subset of Bd[Bd(A)] but I could not prove the converse. Regards, Raed.
7. Topological Space, Finite Intersections

My book gives the following as an exercise: Let (X,T) be any topological space. Verify that the intersection of any finite number of members of T is a member of T. [Hint: to prove this result use "mathematical induction."] Doesn't this follow directly from the definition of a topology? If...
8. Category theory of topological groups

How does one show (directly) that the category of topological groups is protomodular?
9. Topological Space, Continuous, G delta

Let X and Y be topological spaces, and let f:X->Y. Let C denote the set of points at which f is continuous. What axioms (of separation) must X and/or Y satisfy such that C is a G delta set?
10. Definition of convex polytope (I think it's topological problem)

I know, in general, a convex polytope is an intersection of halfspaces described by a system of "inequalities". But what if these inequalities are replaced by equations, namely, a system of equations, just like solving it using algebra? Geometrically is it also a convex polytope by definition...
11. A topological puzzle

Find two sets, P and Q, such that: 1) Both P and Q are contained within the square in R^2 with vertices (1, 1), (1, -1), (-1, -1) and (-1, 1). (In other words all (x, y) in P or Q must satisfy -1\le x\le 1, -1\le y\le 1.) 2) P contains the diagonally opposite points (1, 1) and (-1, -1)...
12. showing that a topological space is t4

Hi there If I can prove that any point in a topology is open (for example the Sorgenfrey line) then it logically follows that the topological space concerned is T4 am I right? Are there other usual ways to show whether a topological space is T4? Regards
13. A question about multiresolution analysis (from a topological point of view)

Hi, I have a problem understanding something This is a snapshot of a book I am reading Point no. 2 concerns me, because it looks to me like it contradicts itself, with "this or this" The first part says \sum_{j}V_j = {L^2(R)} which, to me, looks completely equivavalent to \lim_{j...
14. Countability in topological spaces

Hi, I need to prove the following: Consider the topological space \mathbb{R} with the euclidian topology. Define an equivalence relation on \mathbb{R} as follows xRy \Leftrightarrow x=y \ \mbox{or} \ \{x,y\} \subset \mathbb{N}. Now, consider the quotient topology on \mathbb{R}/ R then this...
15. In the topological space of R, Show that joins do NOT distribute over meets?

I am studying topology for computer science, using a book titled "topology via logic" and the question was an exercise from it. How would you prove that? It seems that the opposite is true. Isn't it?
16. Compactness of topological space

Hey. My assignment says the following: Let X be a topolical space with \tau as its topology. Let \infty be a point not in X. Let X^* = X \cup \{ \infty \} . Let \tau^* = \tau \cup \{ U \in X | X^* \setminus U is a closed, compact subset of X \}. (1) Prove that \tau^* is a topology on...
17. Proof of topological space

I'm new to topology and have just started looking at whether spaces are topologies or not, I've come across a question where i have to show a space is a topology on N and am not sure how to approach it, T = {Tn :n ∊N} ∪ {∅} where Tn ={m ∊N:m ≥ n} I am supposed to prove T is a topology on N...
18. Topological Metric Space with 3 elements

Can you create a topological metric space from three points (the set X) in the plane? Obviously, d is defined. A ball at each point is the point itself (satisfies def of neighborhood of a point). Union of all the balls is X. Intersection of any two balls is empty (contains no member of X) so...
19. property of a topological group??

Hello; Show that if G is a first countable topological group, then there is a sequence of symmetric neighborhoods (U_{n})_{n} of the neutral element e of G such that (1){ {U_{n}:n\in Z }} is a local base at e in G (2) U_{n+1}^{3}\subset U_{n}, for every n I understand number 1 is due to...
20. A is a subset of a topological space X

I am having trouble understanding topology. I have read the sections 3 times and hasn't helped. Could someone explain this question, the methodology to answering it, and how it is done? Let X be a topological space; let A be a subset of X. Suppose that for each x\in A there is an open...