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 (2)

0
votes
1answer
41 views

a bout minimal basic open sets

If f is a continuous function from a finite space to another finite space does the image of minimal basic open set at a point x in the domain is equal to the minimal basic open set at f(x) in the ...
0
votes
1answer
57 views

Interior of subset in product topology = product of each subset in own topology

Let $X$ and $Y$ be two topological spaces. For $A \subset X, B \subset Y$, we consider $A \times B$ as a subset of $X \times Y$. Show that: $\text{Int}(A \times B) = \text{Int}(A) \times ...
0
votes
1answer
53 views

proving a space is a topological space

In terms of Zariski Topology, how would I go about proving $\mathbb{R}^2$ is a topological space? Would that imply showing that the empty set and the whole set are closed, a finite intersection is ...
0
votes
1answer
34 views

Which of the following subsets are dense in the given spaces?

The sets of trignometric polynomials in thespce of continous functions on $[-\pi,\pi]$ which are 2$\pi$ periodic (with the sup norm topology). The subset of $C^{\infty}$ function with compact support ...
0
votes
1answer
80 views

comparison of 3 topologies on C[0,1]

I have a ring of continuous functions from $[0,1]$ to $\Bbb R$. And two norms $C[0,1]\to\Bbb R$. One is supremum of $|f(x)|,$ the other the value of $\int_0^1|f(x)|$. Then I get a Cartesian product of ...
0
votes
1answer
27 views

Question about density of subsets

Let $X$ be a metric space, $U$ an open subset and $A$ a dense subset in $X$. How we prove that $\overline{U} = \overline{U \cap A}$? I think that $\overline{U \cap A}\subset \overline{U}$, what's ...
0
votes
1answer
38 views

X-infinite, consider zariski topology.

Which subsets are closed in X with zariski topology? Is the set of integers closed or open in zariski topology on R? What kind of Union of open intervals is open in zariski topology and Euclidean ...
0
votes
1answer
43 views

Irreducibility of set

Let $Y$ be a subset of the topological space $X$ and let $\{U_i\}$ be an open cover of $X$. 1.If $Y$ is not contained in any $U_i$ then $Y$ is irreducible? 2.If $\{V_j\}$ is an open cover of $Y$ ...
0
votes
1answer
93 views

General Topology - separation axiom

Recall that $a$ is an accumulation point of a set $A$ in a space X if and only if each neighborhood of $a$ meets $A$ in some point other than $a$. We say $a$ is a condensation point of $A$ if and only ...
0
votes
1answer
69 views

Show that every finite-dimensional topological vector subspace is closed.

Let $X$ be a normed topological vector space. Show the following: (i) If $0\neq v \in X$, then $\{\alpha v:\alpha\in \mathbb{R}\}$ is closed. (ii) If $Y$ is a closed vector subspace of $X$ and $w\in ...
0
votes
1answer
42 views

Continuity in terms of nets

Let $f:(X,\tau_X) \to (Y,\tau_Y)$ be a function between two top. spaces. Then $f$ is continuous $\iff$ $\forall Z \subset X: f(\overline Z) \subset \overline{f(Z)}$. I want to prove this in terms of ...
0
votes
1answer
46 views

Relationship between f(X) and f(closure of X)

I am trying to prove if f is continuous and closed ("closed" means the image of any closed subset of the domain is closed) then f(closure of X) equals the closure of f(X). I was able to prove that if ...
0
votes
1answer
31 views

Periodic automorphism of $S^{3}$ and fixed point set

Let $f: S^{3} \rightarrow S^{3} $ be a periodic orientation preserving homeomorphism of order $p$. Suppose there exist two circles $S^{1} $, $A$ and $B$ such that $f(A ) = A $ and $f(B ) = B$. Then is ...
0
votes
1answer
41 views

Condition to become a space $T_1$

A space $X$ is $T_1$ if the diagonal in the product topology $X$ * $X$ is the intersection of family of open sets.
0
votes
1answer
67 views

Relationship between fundamental polygon and its side edges

Here is the fundamental polygon diagram for torus and the diagram for its edge of the square region: My question is why the direction of loops in both circles in the right diagram must be ...
0
votes
1answer
30 views

Support of form and embedded varieties

I need help with some inclusions. Let $i: S \rightarrow M$ be an embedding between two oriented varieties of dimension k and n respectively. Assume that the $i(S)$ is closed and that $\omega\in ...
0
votes
1answer
62 views

Visual intepretation of Topological Properties

I'm studying topology at this moment and i'm wondering if someone know where to find a summary of something of this kind that gives a visual intepretation of the standard topological properties. For ...
0
votes
1answer
33 views

Topology - Metric Space

Show that $d_3$ is a metric space for $\mathbb{R}^2$: $d_3((x_1,y_1), (x_2,y_2))=max.${$|x_1-x_2|,|y_1-y_2|$}$.$ To show something is a metric space d has to have the following properties: (a) ...
0
votes
1answer
132 views

Is the subset [0,1) of $\mathbb{R}$ compact in the lower limit topology?

What I have done so far is give a contradiction, namely the cover: $\mathcal{U}=\{{[0,1-\frac{1}{n}):n\in\mathbb{N}}\}$ Because $\cup_{n\in\mathbb{N}}[0,1-\frac{1}{n})=[0,1)$, it means that there is ...
0
votes
1answer
44 views

Topology - Product Space

Let $\mathbb{R}$ have the usual topology. Describe a subset of $\mathbb{R}^2 = \mathbb{R}\times\mathbb{R}$ that is open in the product space, but that is not a product of open subsets of ...
0
votes
1answer
82 views

Union of intersecting connected sets is connected

$A$ and $B$ are connected sets. If the intersection of $A$ and $B$ is not empty, prove that the subspace $A \cup B$ is connected. My proof: Assume that the union of $A$ and $B$ is not connected, then ...
0
votes
1answer
63 views

Prove that the space of divergent sequences in $(l_{\infty},d_{\infty})$ is open and dense. Is it separable?

The problem statements are: Consider the space $A=\{ \{a_n\}_{n \in \mathbb N} \in l_{\infty} : \{a_n\}_{n \in \mathbb N} \text { is not convergent }\}$ $a)$ Prove that $A$ is open and dense in ...
0
votes
1answer
84 views

The boundary of a bounded open set is not contractible

Suppose we have a bounded open set $\Omega\subset\mathbb{R}^n$: it is well-known that the fact that a retraction of $\overline{\Omega}$ onto its boundary $\partial\Omega$ cannot exist can be directly ...
0
votes
1answer
41 views

prove that the family of the cross product of open sets Oi: Oi is in Bi, i = 1, 2…

Prove: Let $B_1, B_2,\ldots, B_n$ be the bases for topology spaces $(X_1,T_1), (X_2,T_2),\ldots,(X_n,T_n)$, respectively. Then the family $$\{U_1 \times U_2 \times\ldots\times U_n: U_i\in B_i, i = ...
0
votes
1answer
24 views

Action of discrete subgroups E(n) on $\Bbb{R}^n$

Isometry group of euclidean space $\Bbb{R}^n$ is displayed by E(n). We say that a subgroup G of E(n) is discrete if and only if the subspace topology (from E(n)) on G is discrete. If X and Y are ...
0
votes
1answer
48 views

Question about the proof of $S^3/\mathbb{Z}_2 \cong SO(3)$

I'm trying to show $S^3/\mathbb{Z}_2 \cong SO(3)$ completely rigorously. For that purpose I considered three-sphere $S^3$ as a subspace of the ring of quaternions $\mathbb{H}$ and looked into the map ...
0
votes
1answer
82 views

finding an explanation from munkres about quotient topology

I am trying to learn quotient topology from the book of Munkres where he writes that Let X be the closed Unit ball $\{x\times y : x^2+y^2 \le 1\}$ in $\mathbb{R}^2$, and let $X^*$ be the partition of ...
0
votes
1answer
47 views

Normal Space New Properties

Assume $X$ is $T_1$-space. prove that, $X$ is normal if and only if for every closed set $C\subset X$ and open set $U\subset X$ such that $C\subset U$ there exist an open set $V\subset X$ such that ...
0
votes
1answer
56 views

Cauchy sequence with respect to Hausdorff metric

We know that if $(X,d)$ is a complete metric space, then $(CB(X),H)$ is complete too, where $CB(X)$ is the collection of non-empty closed bounded subset of $X$ and $H$ is the Hausdorff metric induced ...
0
votes
1answer
119 views

Discrete subgroups of isometry group $\mathbb{R}^n$

Let $G$ be a Hausdorff topological group. We say that a subgroup $S$ of $G$ is discrete if and only if the subspace topology (from $G$) on $S$ is discrete. Note that isometry group of euclidean space ...
0
votes
1answer
92 views

Continuity of inverse on compact sets.

Problem: The inverse of a continuous injective function $f: A \rightarrow \mathbb{C}$ on a compact domain $A \subset \mathbb{C}$ is also continuous. My attempt: We want to prove that $f^{-1}$ is ...
0
votes
1answer
48 views

Continuity of function $T: \left( C[0,1],d_{\text{sup}} \right) \rightarrow \left( C[0,1],d_{\text{sup}} \right)$

Let $$T: \left( C[0,1],d_{\text{sup}} \right) \rightarrow \left( C[0,1],d_{\text{sup}} \right)$$ where $$d_{\text{sup}}(f,g) = \mbox{sup}_{x \in [0,1]} |f(x) - g(x)|$$ and the definition of $T$ is: ...
0
votes
1answer
15 views

Show $((0,1),\mathcal{U}_{(0,1)})$ and $((0,4),\mathcal{U}_{(0,4)})$ are homeomorphic.

This is what I have: Need to show that $f:(0,1) \rightarrow (0,4)$ given by $f(x)=4x$ is a homeomorphism So I've shown that f is continuous and an open function which is part of the proof, but the ...
0
votes
1answer
50 views

How to enclose a ball more than once with a surface homeomorphic to $S^2$? In 3D.

In 3 dimensional space, how to enclose a monopole (either point-like or ball-like, both types exist.) more than once with a surface homeomorphic to $S^2$?
0
votes
1answer
31 views

prove the existance of $U$ and $V$ Neighbourhoods of $X_0$ and $F(X_0)$ so that $F_{|U}$ is a diffeomorphism in $V$

we have $E = M_n(R)$ and : $$F : E \to E , F(X) = X^2 + X - I$$ we need to prove the existance of $U\in \mathscr V(X_0)$ and $V\in > \mathscr V(0_E)$ so that $F_{|U}$ (restriction ...
0
votes
1answer
27 views

Let $X=\{a,b,c\}$ and $\mathcal{T}=\{X,\emptyset,\{a\},\{b\},\{a,b\}\}$. Let $A=\{a,b\}$ Find each of the following

Let $X=\{a,b,c\}$ and $\mathcal{T}=\{X,\emptyset,\{a\},\{b\},\{a,b\}\}$. Let $A=\{a,b\}$ Find each of the following sets: $Int_{A}(\{a\})$ $Int_{X}(\{a\})$ $Int_{A}(\{c\})$ Can anyone explain how ...
0
votes
1answer
85 views

Intuition of a Submanifold

Could someone explain the intuition behind a submanifold. When, for example, is it appropriate to work with immersed submanifolds vs embedded submanifolds? Why is it important for a submanifold to be ...
0
votes
1answer
45 views

Inverse image of codomain

Is it true that if $f: X \rightarrow Y$ is continuous function then $f^{-1}[Y] = X$ ? I suppose that it is true and I try prove it: Suppose that $f^{-1}[Y] \neq X$ so exsists $x$ such that $ x \in ...
0
votes
1answer
48 views

how can I prove that the rational sequence topology is zero dimensional

I tried to prove that the rational sequence topology is zero dimensional that is X is T1 space and for every x in X , and every closed set C doesnt contain x , there exists a clopen subset U such that ...
0
votes
1answer
54 views

What is the closure of these sets?

I am working through the problems in Topology by Munkres. This comes from Section 17, #17 on page 101. "Consider the lower limit topology on R and the topology given by the basis C, where C = {[a,b) ...
0
votes
1answer
56 views

Klein bottle and real projective two-space

I've been looking at equivalence relations on the unit square: $[0,1] \times [0,1]$ that give rise to various surfaces such as the m$\ddot{\mathrm{o}}$bius strip, but I'm not too sure about the Klein ...
0
votes
1answer
50 views

Verification - Are the intersections/unions of these sets convex/compact/bounded?

I'm looking to see if I have found the correct answers to these questions. Define for $a, b \in \mathbb{R^{2}}, d>0, r>0 $ the sets $V_{a;d} = \{x \in \mathbb{R^{2}}:\max\{|x_{1}-a_{1}|, ...
0
votes
1answer
45 views

Why Urysohn's metrization theorem must hold? And what is its motivation?

Urysohn Metrization Theorem: If $(X\tau)$ is second countable $T_3$, then there exists an imbedding $f:X\rightarrow \mathbb{R}^\omega$. By using Urysohn's Lemma, there exists a countable family of ...
0
votes
1answer
68 views

Do you know a collectionwise normal topological vector space that is not paracompact?

I am looking for an example of a collectionwise normal topological vector space that is not paracompact. Any idea about it?
0
votes
1answer
44 views

Conditions for convergence of sequences in quotient topological spaces

Let $X$ be a topological space and let $\sim$ be an equivalence relation on $X$. Let $X/_\sim$ be the quotient space, endowed with the quotient topology: a subset $\overline{W} \subseteq X/_\sim$ is ...
0
votes
1answer
39 views

first countability and some totally disconnected spaces.

Let $X$ be a topological space that can be written as a countable union of pairwise disjoint clopen subsets. Is $X$ first countable.
0
votes
1answer
27 views

sequential and hereditary Lindelöf

A space $(X, \tau)$ is said to $T_B$ if each compact subset is closed. A space $(X, \tau)$ is said to strongly $T_B$ if each countably compact subset is closed. A space $(X, \tau)$ is said to ...
0
votes
1answer
46 views

Question about Stone-Čech compactification

Assume $R=(\mathcal{U}_n)_n$ is a sequence of distinct ultrafilters on some set $X$. Since every Hausdorff space has an infinite discrete subspace, there is a subsequence $R=(\mathcal{V}_n)_n$ of ...
0
votes
1answer
20 views

Let $A=(0,1)\cup (1,2]$ be a subset of $(\mathbb{R},\mathscr{U})$. Find A' (limit point) and $A \cup A'$

For A' I got $\mathbb{R}-(0,1)\cup (1,2]$ For $A \cup A'$ I got $\mathbb{R}$ But I don't think thats right? Can anyone help me with explaining how to find these?
0
votes
1answer
88 views

Sequence of connected closed subsets of plane?

Suppose $S_1, S_2, S_3, ...$ is a sequence of connected, closed subsets of the plane and $S_1 \supset S_2 \supset ...$ Is $S = \cap S_n$ connected? My reasoning is that S is connected because $S$ = ...