# 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...