Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; ...

learn more… | top users | synonyms (3)

3
votes
2answers
206 views

Riemannian manifolds are metrizable?

I've seen lots of casual claims that Riemannian manifolds (even without assuming second-countability) are metrizable. In the path-connected case, we can use arc-length to create a distance function. ...
6
votes
3answers
341 views

Given an example of a metric space in which every sphere has two centers

This is a question from Wilansky "Topology for analysis", P.15 Prob. 103 Maybe I was thinking too Euclidean, I can't come up some other "centers" of the sphere :(
0
votes
2answers
63 views

How Construct Clopen-Compact Bitopological Spaces?

Dear all who love general topology, In general topology we know the notion of clopen-compact spaces (introduced by A. Sostak): a topological space $(X,\tau)$ is called clopen-compact if every clopen ...
0
votes
2answers
117 views

an example of a continuous bijection which is not a homeomorphism [duplicate]

I need an example of a continuous bijection $f:X \to Y$, where $X$ is NOT compact and $Y$ is Hausdorff, such that $f$ is not a homeomorphism. (It is easy to show that if $X$ is compact, then $f$ is ...
0
votes
1answer
51 views

About the interior ball condition of a convex set with C^1 boundary

Let $\Omega$ an open bounded and convex domain in $R^n$. Suppose that the boundary of this set is $C^1$. Then $\Omega$ satisfies the interior ball condition for all boundary points? Intuitively ...
2
votes
4answers
371 views

The set of points where two continuous functions agree is closed.

I want to prove that if $f,g$ are continuous functions from a topological space $(X,\tau)$ to a metric space $(Y,d)$ then the set $$ A = \{ x \in X : f(x) = g(x) \} $$ is closed. I found a very ...
2
votes
2answers
77 views

Closed Intervals

How do topologists prove continuity of a function with the usual topology at the endpoints of a closed interval? For instance, how would a topologist prove continuity for $f(x)=x^2$ on the closed ...
2
votes
1answer
79 views

If $p:E \to B$ is a covering map, and if $E$ is compact, prove that $p^{-1}(b) $ is finite for all $b \in B$.

If $p:E \to B$ is a covering map, and if $E$ is compact, prove that $p^{-1}(b) $ is finite for all $b \in B$. I need to verify correctness of my proof and ask if there is a more straight-forward ...
0
votes
1answer
92 views

Fundamental group smash product

is there a way to conclude what the first fundamental group of the smash product of two path-connected spaces is? probably there should be a general way like there is for the wedge sum due to van ...
5
votes
0answers
97 views

Properties of first-countable spaces

Hi I have questions regarding first-countable spaces. I just want to confirm something: The following are properties regarding limits and continuity of first countable spaces on Wikipedia: If $f$ ...
4
votes
1answer
104 views

Fundamental group of quotient of $S^1 \times [0,1]$

I have a past qual question here: Let $X = S^1 \times [0,1] /{\sim}$, where $(z,0) \sim (z^4,1)$ for $z \in S^1 = \{ z \in \mathbb{C} \colon \| z \| = 1 \}$. Compute $\pi_1(X)$. I've been trying to ...
0
votes
1answer
66 views

Minkowski Distance Metric

Given compact sets $A$, $B$, define the Minkowski distance between the two sets as: $$ \delta(A,B):= \inf \{ r: B \subseteq \mathscr{N}_r (A) \, \, \text{and} \, \, A \subseteq \mathscr{N}_r (B) \}$$ ...
3
votes
2answers
387 views

Show that two spaces are not homeomorphic

Let $H=[-1,1]\times \{0\}$ and $V=\{0\}\times [-1,0)$ in the plane. Let $T=H \cup V$. Show that $T$ is not homeomorphic to the unit interval $I=[0,1]$. My idea for this problem is that , if we remove ...
2
votes
2answers
129 views

Topology and Arithmetic Progressions

I'm self-studying from "Elementary Topology Problem Textbook" by O.Ya.Viro et al. This is Exercise 2.Lx : Consider the following property of a subset $F$ of the set $\mathbb{N}$ of positive ...
1
vote
1answer
125 views

Show that composition of continuous function is continuous in product topology.

Suppose $H: X \times I \to Y$ is a continuous map of topological spaces $X,Y$ and $I = [0,1]$. And suppose $K: Y \times I \to Z$ is also a continuous map of topological spaces. I want to show that ...
1
vote
1answer
129 views

semirings and basis of a topology

Let $S$ be a semiring of subsets of a nonempty set $X$. What additional requirements must be satisfied for $S$ to be a base for a topology on $X$? Prove that if such is the case, then each member of ...
1
vote
1answer
128 views

Proving properties of closures using intersection of indexed sets and topology

How would I write a proof for this example? Let X be a set, and let $B \subseteq \mathcal P \left({X}\right)$. Define $B^* =${ $U \subseteq X:$ There is an index set $I$ and $U_{i} \in B$ for each ...
3
votes
0answers
34 views

Ultrametric space of stochastic filtration

Let $\Omega$ be an arbitrary set and $(\mathscr F_t)_{t\in \Bbb R_+}$ be a non-decresing sequence of $\sigma$-algebras on $\Omega$ such that any subset of $\Omega$ is contained is some of them, that ...
5
votes
1answer
78 views

If $\{\tau_\alpha\}$ is a family of topologies on $X$, show that $\cap \tau_\alpha$ is a topology on $X$. Is $\cup \tau_\alpha$ a topology on $X$?

If $\{\tau_\alpha\}$ is a family of topologies on $X$, show that $\cap \tau_\alpha$ is a topology on $X$. Is $\cup \tau_\alpha$ a topology on $X$? For all $\alpha$, $\varnothing \in \tau_\alpha$ ...
3
votes
1answer
113 views

Is the collection $\tau_\infty = \{U:X-U$ is infinite or empty or all of $X\}$ a topology on $X$?

Can someone please verify my proof? Is the collection $\tau_\infty = \{U:X-U$ is infinite or empty or all of $X\}$ a topology on $X$? No. Let $X = \mathbb{R}$. Clearly, $\{x\} \in \tau_\infty$ ...
0
votes
1answer
65 views

Rudin Real and Complex Ch.2 question 16

This excerise 2.16 in Rudin is as follows: Let X be the plane with the following topology: a set is open iff it's intersection with every vertical line is an open subset of that line w/ respect to the ...
2
votes
2answers
40 views

Prove that regular $T_1$ space is $T_2$ space.

Prove that regular $T_1$ space is $T_2$ space. Definition of $T_1$: For all $a,b\in X$, there exist $A,B\in\tau$ s,t, $a\in A, b\notin A,b\in B,a\notin B$. Definition of regular: For all $A\in ...
0
votes
1answer
53 views

Question about proofs with topological spaces

How would I write a proof for this example? Let $(X, \tau_{1})$, $(Y, \tau_{2})$ and $(Z, \tau_{3})$ be topological spaces. A function ${f}: X \rightarrow Y$ is said to be continuous if for every V ...
1
vote
1answer
46 views

nonempty interiors can't be defined by their infinite behavior

Show that there is no topology with the property that the interior of any set is nonempty if and only if the set is infinite.
4
votes
1answer
70 views

Homotopy classes of maps from the projective plane to $S^1 \times S^3$

I have a past qual question here: characterize the space $[(\mathbb{RP}^2,x),(S^1 \times S^3,y)]$ of homotopy classes of maps from $(\mathbb{RP}^2,x)$ to $(S^1 \times S^3,y)$, where here $x \in ...
0
votes
1answer
20 views

Question about Neighborhood basis

In the Simon Reed text, after defining the strong operator topology it is said: "A neighborhood basis at the origin is given by the sets of the form $\{S \ | \ S \in \mathcal{L}(X,Y), ...
3
votes
2answers
61 views

The union of all the open sets in a family of topologies

I'm starting studying topology for the first time and my teacher just wrote this. I just don't understand the last line: Let $\{\tau_\alpha\}$ be a family of topologies on X. [...] To say that ...
2
votes
2answers
135 views

intersection number of twocompact oriented manifolds

I have an oriented manifold M of n dimension and 2 oriented submanifolds, one of dimension k and the other of dimension n-k , I have to understand which is the intersection number of those manifolds. ...
0
votes
1answer
134 views

Homology of 3-sphere minus an embedding of $S^1 \times \mathbb{D}^2$

I'm having trouble with the following past qual question: Let $\phi \colon S^1 \times \mathbb{D}^2 \hookrightarrow S^3$ be an embedding, where $\mathbb{D}^2$ is the open unit disk in $\mathbb{R}^2$. ...
0
votes
1answer
50 views

Linear bijection non-preserving Hausdorff propery

My question is: If $f: X \to Y$ is a continuous and linear bijection between topological vector spaces, is it possible that $X$ is Hausdorff and $Y$ is non-Hausdorff? (TVSs are considered in the more ...
2
votes
1answer
84 views

The definition of Compactness for “set” and “space”

Compactness for "set" and "space" I was wondering if there is any significance between the two settings. Do we treat them as two different things? For example, let $(X,d)$ be a metric space with the ...
1
vote
0answers
153 views

If $X$ is compact and $f:X \rightarrow Y$ is a dense continuous injection, then $f$ is a homeomorphism

I found this: Let $X$ be a compact space and $f:X \rightarrow Y$ a continuous injection. Let $f(X)$ be dense in $Y$. Prove that $f$ is a homeomorphism. So, my question is: is it possible to prove ...
0
votes
0answers
69 views

Link complement simply-connected if codimension $\geq 3$

In Rolfsen, page 50 says that "an easy general position argument shows that a PL link $L^k$ in $S^n$ has simply-connected complement if $n - k > 3$," where $L^k$ is a $k$-dimensional link in $S^n$. ...
8
votes
2answers
140 views

A metric on $\mathbb{N}$

Define a metric on $\mathbb{N}$ by fixing a prime, $p$, and setting $$d(x,y)=\begin{cases} 0 & x=y \\ p^{-k} & \text{otherwise} \end{cases}$$ where $p^k$ is the highest power of $p$ that ...
3
votes
2answers
227 views

A question on the proof of 14 distinct sets can be formed by complementation and closure

In Munkres, problem 20 of Section 2-6, it says that 14 distinct sets can be formed by complementation and closure. I see only five so far. Let f be the function of closure mapping and g be the ...
2
votes
1answer
75 views

Group action and set define via their quotient topology open/closed equivalence relations

If we have a topological space $X$ and a subset $A \subset X$, we can define $X \backslash A$. My question is: Is it true that this equivalence relation is closed iff $A$ is closed as a subset of $X$ ...
3
votes
1answer
100 views

How to show a set is compact in a function space?

I have a question asking if $\{f_n\}$ is a compact in $C_b([0,\infty))$ (bounded continuous) with $||\cdot||_{L^\infty}$. The sequence is $$f_n (t) = \sin\sqrt{t+(2n\pi)^2},$$ I have showed that ...
1
vote
0answers
25 views

Non-section representation of an intersection of sets

Let $X,\bar X,Y$ be arbitrary sets and $A\subseteq X\times Y$, $\bar A\subseteq \bar X\times Y$ be arbitrary as well. Denote: $$ A_x :=\{y\in Y:(x,y)\in A\} $$ and similarly for $\bar A$. Consider a ...
0
votes
1answer
359 views

Fundamental polygon

So, I have seen fundamental polygons quite a few times now and I was always wondering what they are actually good for. Let's take the sphere. It's fundamental polygon can be seen here image. Does ...
3
votes
4answers
577 views

A set of all rational numbers in $[0, 1]$?

I have a question that is giving me some minor grief: If $A$ is a closed set containing all rational numbers $r \in [0, 1]$, then show that $[0, 1] \subset A$. I don't really understand this ...
-2
votes
1answer
161 views

Prove that intersection of connected spaces is connceted.

Let A and B be connected subspaces of a topological space (X,$\tau$). If A,B are not disjoint, prove that the subspace A $\cap$ B is connected. Using the definition of connected space is that the ...
7
votes
1answer
78 views

Step Connected if and only if Connected

A space $X$ is step connected if given any open covering $\mathcal{U}$ of $X$ and any pair of points $p,q\in X$ there is a finite sequence $U_1,\ldots,U_n$ of sets belonging to $\mathcal{U}$ so that ...
1
vote
0answers
47 views

Completeness of Locally Compact Metric Space and Group of Isometries

Let $X$ be a locally compact metric space, and suppose that the group of isometries of $X$ acts transitively. Show that $X$ is complete. (This is 2nd part of a problem. In first part I showed that for ...
1
vote
2answers
276 views

A proof that if the product of spaces is Hausdorff, each of them is Hausdorff

Is my approach to this question right? Question: Prove that if $$\prod_{\alpha \in J} X_\alpha (\neq \emptyset) $$ is Hausdorff, each $X_\alpha$ is Hausdorff. Attempt to answer: It is enough to ...
2
votes
2answers
405 views

Parametric formula for figure 8 mobius strip

I'm making 3D prints with Mathematica, and am interested in a parametric formula for a mobius strip that is in the form of a figure 8, rather than simply a circle with a twist in it. Can someone help ...
2
votes
0answers
87 views

A Homeomorphism that is not unique even upto Isotopy

I'm currently reading the following paper by Richard Skora, entitled Cantor sets in $S^3$ with simply connected complements found here, and on page 2, just before Theorem 1, it says "the homeomorphism ...
1
vote
0answers
32 views

Mapping open on open dense subset => Mapping is open on whole space?

Let $X,Y$ be topological spaces, and let $f\colon X \to Y$ be a continuous function. Further suppose that there exist an open and dense subset $U$ of $X$, such that $f\vert_{U} \colon U \to Y$ is an ...
0
votes
1answer
54 views

Does the boundaries of non-disjoint sets in Euclidean space have common element?

I've got stuck while solving a problem, and the thing I need is; If you are given two open sets in $\mathbb{R}^{n}$, where they have both common element and non-common element. (That means, their ...
0
votes
1answer
42 views

If $A=[0, 1] \times (0, 1)$, which is a subspace of $I^2 = [0, 1] \times [0, 1],$ how are the sets $U_x = \{x\} \times (0, 1)$ open in $A$?

If each of $U_x$ is open, doesn't this imply that $\{x\}$ is open in $[0, 1]$, which contradicts the uncountability of $[0, 1]$? This question arises from an example (#5) in James Munkres' Topology, ...
4
votes
3answers
441 views

If two Borel measures coincide on all open sets, are they equal?

Let $X$ be a topological space and let $\mathcal{B}(X)$ be its Borel $\sigma$-algebra. That is, $\mathcal{B}(X)$ is the smallest $\sigma$-algebra on $X$ containing all the open sets. Let $\mu, \eta : ...