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
1answer
24 views

Is $X = \{0,1,…,9\}^{\mathbb N}$ totally disconnected?

Consider X with the product topology and $\{0,1,...9\}$ with the discrete topopogy. I already attempted to prove and disprove it, both without success, so I'd be happy to know whether it is true or ...
0
votes
1answer
6 views

finding $C^1$ path on an open and path connected set.

Given an open and path connected set $U\subseteq \mathbb R^n$, is there a way to find a $C^1$ path between every $a,b\in U$? If so, is there a general proof of existence of such path?
-1
votes
0answers
25 views

semicontinious functions in topology space [on hold]

Problem is: Let $X$ be a topologicyl space.. Prove that: A function $f\colon X\to\mathbb R$ is continuous function if and only if $f$ is lower semi-continuous and upper semi-continuous. The ...
0
votes
1answer
8 views

Nets and sequences in a 1st couuntable space.

Let $(X,\mathcal{T}$) be a 1st countable topological space. Let $(x_\delta)_{\delta\in\Delta}$ be a net converging to $x$. Does there exist a sequence $(x_n)$ that converges to $x$ and which is a ...
1
vote
0answers
31 views

Proving a function is continous only if a set is open

FACT: A function is continuous if and only if the inverse image of every open set is open. Now, suppose we have $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous everywhere and lets say we have an ...
6
votes
3answers
164 views

If a set is compact then it is closed

Show that if a set is compact then it is closed. definitions: Let $A\subset \mathbb{R}$. A point $p\in\mathbb{R}$ is an accumulation point or limit point of $A$ if and only if every open set $G$ ...
3
votes
1answer
20 views

Difference between continuous and uniformly continuous functions on a dense metric subspace.

Let $X$ be a dense subset of metric space $(\tilde X,d)$. Let be $(Y,d')$ be a complete metric space and $ f: X \rightarrow Y$ a continuous mapping. It follows from density that for all points in ...
0
votes
0answers
29 views

Is $\{\frac{1}{n}:n\in\mathbb{N}\}$ nowhere dense in $[0,1]$? [duplicate]

Is $\{\frac{1}{n}:n\in\mathbb{N}\}$ nowhere dense in $[0,1]$ for the metric induced from the Euclidean metric on $\mathbb{R}$? I think that yes, it is nowehere dense because ...
0
votes
0answers
16 views

Interpretation of a weakly compact set of functions

I'm having trouble really grasping the idea of a weakly compact set. The set I have under consideration is a set of functions $M_c$, where $$M_c=\{f:W(f)\leq c\},$$ where $c\geq0$ and $W(f)$ is a ...
0
votes
0answers
35 views

Do finite products commute with colimits in the category of spaces?

Let $X$ be a topological space. The endofunctor $\_\times X$ of the category of all topological spaces does in general not possess a right adjoint, since the category is not cartesian closed. Is it ...
-1
votes
0answers
14 views

Which of the following are proper patches. (Showing that an inverse of a mapping is continuous)

In which of the following cases is the mapping $\mathbf{x}:\mathbb{R^2} \to \mathbb{R^3}$ a proper patch? (a)$\mathbf{x}(u,v)=(u,uv,v)$ (b)$\mathbf{x}(u,v)=(u^2,u^3,v)$ ...
0
votes
1answer
14 views

open set in subspace

Could someone check the following? Consider $M=\{0\} \cup \{ 2^{-k}: k \in \mathbb{N}\}$ a subspace of $\mathbb{R}$. Which of the subsets $\{0\}$ and $\{2^{-k}\} (k\in \mathbb{N})$ are open in ...
1
vote
1answer
13 views

Relation between open divisible subgroup and the quotient of the group with subgroup

I wanted to prove the following proposition: Let H be an open divisible subgroup of an abelian topological group G. Then G is topologically isomorphic to H x G/H. As for the proof, using extension of ...
2
votes
2answers
29 views

a simple question about topological space

X is a topological space, $ A\subseteq B\subseteq X $, if $A$ is a nowhere dense subset of $B$ , then $A$ is a nowhere dense subset of the whole space $X$? Is this right? I thind it's right, but I ...
0
votes
0answers
23 views

Every point in a Tychonoff Space is contained in a compact set

This should be very elementary, but I just can't see it: Is every point in a Tychonoff Space contained in a compact set? I tried to look for a counterexample and figured that it cannot be locally ...
1
vote
1answer
23 views

Closedness of $\{ x \in 2^A : x(\neg p) = \neg x(p) \}$ for a Boolean algebra $A$ and $p \in A$

I'm reading Matthew Dirk's The Stone Representation Theorem for Boolean Algebras, and am trying to follow the proof of Proposition 3.4 on p.6: Proposition 3.4. Let $A$ be a Boolean algebra, and ...
2
votes
1answer
33 views

Number of Hausdorff topologies on a set with $100$ elements.

Find the number of Hausdorff topologies on a set with $100$ elements. I know that number of topologies on a set with $2$ elements is $4$, with $3$ elements is $29$ , with $4$elements is $355$ , ...
0
votes
1answer
32 views

Connected Sets Examples

(a) Give an example of a connected set $A \subset \Bbb R^n$ such that $\Bbb R^n\setminus A$ is not connected. (b) Give an example of a compact set $K \subset \Bbb R^n$ which is not connected. So far ...
1
vote
1answer
28 views

Projection Mappings are Quotient Mappings?

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Prove that if $X=X_1\times X_2$ is a product space, then the first coordinate projection is a ...
2
votes
0answers
25 views

When does the quotient metric is equivalent to the quotient topology?

Suppose that we have an equivalence relation $\sim$ in a topological metrizable space $(X,d).$ Then we can endow $X/\sim$ with the quotient topolgy. Also, under certains circunstances, there exists a ...
5
votes
4answers
72 views

Prove that if $A$ is both open and closed, $A=\mathbb R$. [duplicate]

Suppose $A$ is a non-empty subset of $\mathbb R$. Prove that if $A$ is both open and closed, $A=\mathbb R$. I think I'm supposed to assume that $A$ is not equal to $\mathbb R$ and derive a ...
0
votes
0answers
17 views

Necessary and sufficient condition for an orthonormal system to be total

Let $H$ be a Hilbert space over a field $\mathbb K$. Prove that an orthonormal system $\{a_n\}_{n=1}^{\infty}$ in $H$ is total if and only if: $\forall$ $x \in H$, the following holds: ...
2
votes
1answer
20 views

Image of continuous injective map has empty interior.

Let $\varphi :\left [ 0,1 \right ]\rightarrow \mathbb{R^2}$ be a continuous injective map. Let $I = \varphi \left ( \left [ 0,1 \right ] \right )$ be the image of this map. Prove that $I$ has empty ...
1
vote
1answer
23 views

Dense sets and Empty Interior

if $A$ is dense in $X$, is there a relation which shows in which cases $A$ has empty interior ? $\mathbb{Q}$ has an empty interior as a dense set in $\mathbb{R}$, so does its complementary in ...
0
votes
0answers
17 views

A question on product space [duplicate]

If $|X|=\mathfrak c$, then what is the cardinality of the product space $X^{\omega}$? Thanks very much.
2
votes
1answer
28 views

Example 5, Sec. 23 in Munkres' TOPOLOGY, 2nd edition: What is the closure of this set?

What is the closure in $\mathbb{R}^2$ of the set $$ \left\{ \ x \times y \ \in \mathbb{R} \times \mathbb{R} \ \colon \ x > 0, \ y = \frac{1}{x} \ \right\}? $$ I know that each point of the set is ...
0
votes
1answer
21 views

Example 4, Sec. 22 in Munkres' TOPOLOGY, 2nd edition: How to figure out saturated open sets?

Let $X$ be the closed unit ball $$ \{ \ x \times y \ \colon \ x^2 + y^2 \leq 1 \ \} $$ in $\mathbb{R}^2$, and let $X^*$ be the partition of $X$ consisting of all the one-point sets $\{ \ x \times y ...
1
vote
1answer
41 views

Find two disjoint open sets $U, V$ such that $A\subseteq U, B\subseteq V$ where $A,B$ are closed.

Let $A, B$ be two disjoint closed subsets of a certain metric space $(M,d)$. Show that there exist disjoint open subsets $U, V \subseteq M$ such that $A\subseteq U, B\subseteq V$. Give ...
0
votes
1answer
26 views

Prob. 3, Sec. 22 in Munkres' TOPOLOGY, 2nd edition: How is this map a quotient map that is neither open nor closed?

Let $\pi_1 \colon \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ be projection on the first coordinate. Let $A$ be the subspace of $\mathbb{R} \times \mathbb{R}$ consisting of all points $x \times y$ ...
0
votes
1answer
28 views

I think $A,B$ must be closed and disjoint

Prove that in every metric space, $(X; d)$, is possible find a continous function$f\colon X\to \mathbb{R}$ such, if $ A $ and $ B $ are two subsets of $ X $ then $ f(x) = 1 $, for every $ x\in A $ ...
1
vote
0answers
15 views

Prove that the “additive” operation of the module($\mathbb{Z}_{p}^{*},\mathbb{Z}_{p-1},\cdot$) is continuous.

Consider the following module $\mathcal{M}=$($\mathbb{Z}_{p}^{*},\mathbb{Z}_{p-1},\cdot$) in which the "additive" operation is defined by normal multiplication in $\mathbb{Z}_{p}^{*}$ and scalar ...
2
votes
1answer
11 views

Hausdorff spaces for continuous bijections

I have the following question being posed: Suppose $f:X\rightarrow Y$ is a continuous bijection. Prove that if $Y$ is Hausdorff, then $X$ is also Hausdorff. Here's my attempt: Consider any $a,b\in ...
0
votes
1answer
24 views

Find a suitable counterexample?

Is the following statement true or false? If a sequence $(x_n)$ with an infinite range $\{ x_n : n \in \mathbb{N} \}$ has precisely one accumulation point, then $(x_n)$ converges. I know the ...
1
vote
3answers
28 views

Show that the closed ball is closed in $\mathbb{R}^p$

Let $r>0, p \in \mathbb{N}$ be given. Show in detail that the closed ball $\{ x \in \mathbb{R}^p : ||x|| \leq r \}$ is closed in $\mathbb{R}^p$. Let $A = \{ x \in \mathbb{R}^p : ||x|| \leq r ...
-8
votes
0answers
45 views

Id proof this quastion [on hold]

Id proof this quastion pleace tomorw to me
2
votes
2answers
22 views

Examples of path component maps

I understand what needs to be done for the first part, i have to somehow map $1$ point onto $1$ point, in a map where there exists $2$ points... so the inverse map is injective, but how is this ...
4
votes
3answers
77 views

Mistake in (Baby) Do Carmo? Elementary topology of surfaces.

If you have the book, it's proposition 2 of section 5.3. If not, the proposition reads: Given any two points p and q $\in$ a regular, connected surface S, there exists a parameterized piecewise ...
2
votes
0answers
16 views

$\operatorname{Fr}( p(\overline U) )$ where $p : X \to Y$ is a closed, not necessarily continuous, surjection, and $U \subset X$ is open

Question: Let $p : X \rightarrow Y$ be a closed (not necessarily continuous) surjection. If $U$ is open, then $$\operatorname{Fr} ( p(\overline U) ) \subset p(\overline U ) \cap p(X - U).$$ I ...
2
votes
2answers
54 views

Example 1, Sec. 22 in Munkres' TOPOLOGY, 2nd edition: How to verify that this map is closed?

Let $X$ be the subspace $[0,1] \cup [2,3]$ of $\mathbb{R}$, and let $Y$ be the subspace $[0,2]$ of $\mathbb{R}$. The map $p \colon X \to Y$ defined by $$ p(x) \colon= \begin{cases} x \ &\mbox{ ...
1
vote
0answers
17 views

Convex open neighborhood of compact convex subset

I'm stuck on what ought to be a straightforward topology problem. Say $X$ is a compact convex subset of a locally convex space (everything in sight is assumed Hausdorff). Say $Y\subseteq X$ is a ...
1
vote
0answers
12 views

Finite intersection of arbitrary union not stable for arbitrary unions

It is a set-theoretic exercise to prove that the set of arbitrary unions of finite intersections of sets is still stable under finite intersections. However it is not true that finite intersection of ...
1
vote
1answer
20 views

A proof related to diameter of a simplex S

Question: Prove that the diameter $\mathcal p(S)$ of a simplex $\mathcal S$ equals the greatest Eucledian distance between two vectors in the simplex. My opinion: We all know what every vector in the ...
1
vote
0answers
11 views

Existence of real valued function continuous at $\mathbb Q$ discontinuous at $\mathbb R\backslash \mathbb Q$ [duplicate]

Does there exist a real-valued function of a real variable which is continuous at every rational point and discontinuous at every irrational point?
2
votes
1answer
61 views

Section 22 in Munkres' TOPOLOGY, 2nd edition: How to establish this equivalence?

Let $X$ and $Y$ be topological spaces; let $p \colon X \to Y$ be a surjective map. Then $p$ is said to be a quotient map provided a subset $U$ of $Y$ is open in $Y$ if and only if $p^{-1}(U)$ is ...
0
votes
0answers
25 views

Prob. 4, Sec. 21 in Munkres' TOPOLOGY, 2nd ed: How to decide which cases to consider?

We need to show that the ordered square satisfies the first countability axiom. I'm not able to decide as to which separate cases to consider. By definition the ordered square is the product $I ...
0
votes
1answer
38 views

Is a limit point compact subspace of a Hausdorff space necessarily closed? [on hold]

I think the answer should be "no", but I can't give a counter-example.
0
votes
0answers
35 views

If $d(x_0,y_j)\to d(x_0,y_0)$, then $y_j \to y_0$.

Consider a metric space $X$, and a compact subset $C\subset X$.Let $x_0\in X-C$. We can show that there is a point $y_0\in C$ such that $d(x_0,y)=\inf_{y\in C} d(x_0,y)$. Now suppose there is ...
0
votes
1answer
29 views

generalize the question every every intersection of nested sequence of compact non-empty sets is compact and non-empty

I'm aware how to prove that the intersection of nested sequence of compact non-empty sets is compact and non-empty. but I want to generalize this question to transfer the hypothesis of having nested ...
3
votes
2answers
43 views

Is a topological space $X$ the colimit of an open cover $\cup U_i$ in this way?

Let $X$ be a topological space space and $X=\cup_{i\in I} U_i$ a covering of $X$ by open subsets $U_i\subseteq X$. Is it true that $$ \operatorname{colim}\left(\coprod_{(i,j)\in I\times I} ...
1
vote
0answers
19 views

weak closure of unitary group in $B(H)$

Let $H$ be a Hilbert space with dim $H=\infty$ , and $\cal{U}$ be the group of all unitaries on $H$. Show that the weak closure of $\cal{U}$ is a semigroup with respect to the multiplication. I know ...