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; ...

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1answer
25 views

Convergence of filters in topological spaces

I'm having quite some trouble proving the following: 1) Let $X$ be a topological space. If any filter on $X$ converges to any point $x$ $\in$ $X$, show that $X$ is endowed with the trivial topology ...
1
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2answers
45 views

Generalizing limits of sums, products, and quotients of sequences to abstract topological spaces?

Introductory real analysis books usually state some properties about limits of sequences of numbers. Suppose $\lim x_n=x$ and $\lim y_n=y$. Then: $\lim (x_n+y_n) = x+y$ $\lim c x_n = cx$, for $c \in ...
2
votes
1answer
23 views

If $X$ is compact and $C$ is an open finite cover of $X$ then $C$ has a maximum Lebesgue number.

Prove the following statement. If $X$ is compact and $C = \{U_\alpha : \alpha \in A\}$ is an open finite cover of $X$ then $C$ has a maximum Lebesgue number. Is my proof correct? Proof: Let $E$ be ...
0
votes
0answers
74 views

If $\bar X$ is open, then $X=\bar X$.

Let $X$ a metric space and suppose that $\bar X$ is open. Suppose that $\bar X\neq X$. Let $x\in \bar X\setminus X$. By definition of $\bar X$ there is a sequence $(x_n)_{n\in\mathbb N}$ in $X$ such ...
7
votes
1answer
137 views

Topological idea of orientability of manifold

While reading Poincare Duality a new idea of orientability of manifold came in my mind.I dont know wheather this idea is new or not, or even true or false. My idea is following... A $n$-dim manifold ...
2
votes
2answers
55 views

Whether the set $A$ , $ A=\{y\in R : y= \lim_{n\rightarrow \infty} f(x_{n}) \text{ where } x_n \text{ diverges to } +\infty \}$ is connected

Let $f: \mathbb R \rightarrow \mathbb R$ be continuous function and $A\subset \mathbb R$ be defined by $$ A=\{y\in R : y= \lim_{n\rightarrow \infty} f(x_{n}) \text{ where } x_n \text{ diverges ...
3
votes
3answers
44 views

Is $E$ path connected $\implies \overline{E}$ connected?

Let $E\subset \mathbb R^n$ a path connectedness open set. Is $\overline{E}$ connected ? (where $\overline{E}$ is the closure of $E$). I tried to prove that it's true, but I don't get anything, may be ...
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votes
1answer
46 views

How can I prove that $(X,τ)$ is a Hausdorff topological space?

Let $(X_1,τ_1)$ is a Hausdorff topological space and $(X_2,τ_2)$ is a Hausdorff topological space and $X=X_1\times X_2$ and $τ$ The product topology How can I prove that $(X,τ)$ is a Hausdorff ...
3
votes
3answers
56 views

Trying to Understand a Remark about Zariski Topology

I'm reading some notes in which following remark is given: The Zariski topology is quite different from the usual ones. For example, on affine space $ \mathbb A^n$ a closed subset that is not ...
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0answers
32 views

Topology of metric space

If $C[a,b]$ is the set of all real valued continuous functions defined on $[a,b]$ and $(C, d)$ is a metric space where $d(x,y)=\max | x(t)-y(t)|$ and $t$ belongs to $[a,b]$. Then how can i determine ...
1
vote
2answers
22 views

$M$ is open in $Y$ and $M$ is open in $Z$ then $M$ is open in $X$

Is it true? $X= Y \cup Z$, $M$ is a subset of $Y \cap Z$. Suppose that $M$ is open in $Y$ and $M$ is open in $Z$ then $M$ is open in $X$.
2
votes
1answer
43 views

Topology of Metric Spaces

Why is the open interval $(-\infty,+\infty)$ not an open sphere with usual metric? We can find a radius such that the open sphere is subset of real line same as we find that for any open interval. ...
15
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3answers
2k views

Normal subgroups of free groups: finitely generated $\implies$ finite index.

I am looking at what should be a simple exercise in geometric group theory. I have reduced the problem to just completing an exercise from Hatcher, Section 1.B page 87: 7. If $F$ is a finitely ...
1
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0answers
20 views

Quotient Space $X^*$ is homeomorphic to the Subspace $S^2$ of $\mathbb R^3$

Let $X$ be the closed unit ball $\{ x^2 + y^2 \leq 1 \}$ in $\mathbb R^2$ and let $X^*$ be the partition of $X$ consisitingof all the one point set $\{ x \times y \}$ for which $x^2 + y^2 < ...
0
votes
2answers
38 views

Does this theorem for bases also hold for subbases?

Assume that we have a toological space $X$ with toplogy $\mathcal{T}$. If Y is a subspace of X, then $\mathcal{T}_Y=\{Y\cap U|U \in \mathcal{T}\}$ is a topology on Y (that it really is a topology, ...
2
votes
2answers
67 views

For $l^2 (\mathbb{N})$ is $(l^2, d_2)= (l^2, d)$ topologically, where $d$ forms the usual product topology on $\mathbb{R}^{\mathbb{N}}$?

If we define $d(x,y)= \max_{n \in \mathbb{N}}( \min \{2^{-n}, |x_n - y_n| \})$ then does this distance form the same topology on $l^2 ( \mathbb{N})$ (the set of all square-summable real sequences) as ...
2
votes
2answers
46 views

Does this set tend towards a disc?

Let $p$ be a complex polynomial \begin{gather*} p:\mathbb{C}\longrightarrow\mathbb{C},\\ \deg p = n,\quad n\in\mathbb{N}. \end{gather*} Define the set $\mathcal{R}=\{z\in\mathbb{C}:|p(z)|\leq R\}$, ...
0
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1answer
45 views

Every convex set in $\mathbb R^n$ has a countable and dense subset?

Assume the space is Euclidean space. Why every convex set has a countable and dense subset? How about in metric space? Any ideas or references? It is used in process of proving Debreu's Theorem in ...
2
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0answers
36 views

How to construct a ring $ R$ such that $(Spec(R), \tau)$ is not a Sequential Space where $\tau$ is the Zariski Topology on $R$

How to construct a ring $ R$ such that $(Spec(R), \tau)$ is not a Sequential Space where $\tau$ is the Zariski Topology on Spec(R). I've just learned about Zariski topology so I really don't have ...
1
vote
2answers
81 views

Characterization of the weak topology

In our functional analysis lecture we defined the weak topology in a what seems to me like a non canonical way, i.e. not as unions of finite intersections of preimages of open sets in the underlying ...
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0answers
47 views

Determining the interior of $([-1, 1]\times[-1, 1])\setminus \{ y \in \mathbb{R}^2 : d((0, 0), y) < 0.25 \} \subseteq \mathbb{R}^2$

Let $M = (\mathbb{R}^2, d_e)$ be the metric space, with $d_e$ the Euclidean metric. Let $C \subseteq \mathbb{R}^2$ be defined by $$C = ([-1, 1]\times[-1, 1]) \setminus \{ y \in \mathbb{R}^2 : d((0, ...
3
votes
0answers
50 views

Homeomorphism definition: why $f^{-1}$ and not another function?

If you have two topological spaces $X$, $Y$ and two continuous bijections $$g: X \to Y $$ $$f : Y \to X $$ then are $X$ and $Y$ homeomorphic? If not, is there a reason why the above does not serve as ...
3
votes
1answer
22 views

An example of open closed continuous image of $T_2$-space that is not $T_2$

Engelking in his "General Topology" states that $T_2$ separation axiom is not preserved under open closed continuous surjections. In "General Topology" by Stephen Willard I have found two separate ...
3
votes
2answers
45 views

Is the graph of $xy=1$ in $\mathbb C^{2}$ connected?

The graph of $xy=1$ in $\mathbb C^{2}$ is set of points $(x+iy,u+iv)$ that satisfies $$xu-yv=1$$ and $$uy+xv=0$$ How to find if this set is connected or not . I also have another ...
2
votes
1answer
30 views

Two disjoint compact sets in a topological group

Let $(G, \cdot )$ be a compact (Hausdorff) topological group. If $A$ and $B$ are two disjoint compact subsets of $G$, how can we show that there exists a nonempty open set $V$ such $A\cdot ...
4
votes
1answer
116 views

Is $\overline{D}_{\varepsilon}$ a connected Jordan region in $\mathbb{R}^{n}?$

Definition. Let $E$ be a nonempty subset of $\mathbb{R}^{n}$.The distance from a point $\mathbb{x}\in\mathbb{R}^{n}$ to set $E$ is defined by ...
2
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0answers
24 views

decomposing a function into embedding and projection

I have a simple question. If $f:\mathbb{S}^{2}\rightarrow\mathbb{R}$ is a non-constant continuous function, can we represent it as a composition $f=p\varphi$, where ...
1
vote
2answers
14 views

Bounded sequence in a metric space

I have a small question when we have a bouded sequence in a metric space; we say that there exists a closed ball $B'$ such that $(x_n)\subset B'$ or just there exist a ball $B$ such that $(x_n)\subset ...
0
votes
1answer
30 views

Unclear about the definition of “closed”?

A number of resources online cite the definition of "closed" as a set containing all its limit points. But this statements seems to be always true to me. A limit point is one having at least one ...
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votes
1answer
19 views

If $X$ is a polish space, how do we find an equivalent metric under wich $X$ is a totally bounded?

According to Stroock and Varadhan, If $X$ is a polish space, then one can choose an equivalent metric under which the space is totally bounded (see Stroock and Varadhan - Multidimensional diffusion ...
2
votes
3answers
70 views

What is the Quotient on the Coproduct in Adjunction Spaces

Can someone please provide a detailed explanation of the equivalence relation used to construct adjunction spaces from the topological coproduct? In particular, most sources talk about "identifying an ...
2
votes
2answers
46 views

Question about vector spaces with the discrete topology

Is it true that every vector space with the discrete topology is a topological vector space? (That is, a topological space with continuous addition and scalar multiplication whose singletons are ...
1
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2answers
35 views

Characterizing uncountable connected topological spaces

We know that if $X$ is a connected metric space with more than one point , then $X$ is uncountable ; can we characterize those connected topological spaces for which more than one point implies ...
2
votes
3answers
34 views

Is the intersection of the following closed and open set closed? Generally?

Ok, I have been informed that the below lemma is incorrect. I needed it to prove the following statement. Could someone else provide a proof? Statement: If m(E) is finite, there exists a compact set ...
2
votes
3answers
479 views

sequentially continuous on a non first-countable

Can you give me an example of a function which is sequentially continuous but not continuous? (I know that in first-countable spaces this is equivalent, but what about in spaces without this ...
0
votes
2answers
41 views

Can you explain this proof about the closure of a set?

The author of my textbook has an unsatisfactory proof when it is describing the properties of the closure of a set. I'm using $E^*$ for E closure. Also, $E'$ indicates the set of limit points of $E$. ...
1
vote
5answers
90 views

Spaces $X$ in which every subset is either open or closed, and only $\varnothing$ and $X$ are clopen

Let $(X, \tau)$ be a topological space. Then $X, \varnothing \in \tau$ and are both clopen. But I wonder if it is possible to construct a topological space $X$ in which all subsets are either open or ...
9
votes
1answer
63 views

Let $A$ be an open set of $\mathbb{R}$ and $B$ any set, under what coniditions of $B$, $AB$ is open?

I don't really know how to establish the conditions so $AB$ can be open. The problem says: Let $A$ be an open set in $\Bbb R$ and $B$ any other set. Define: $$AB = \{xy\in\mathbb{R}\,\colon x\in ...
2
votes
0answers
50 views

Give an example of a function $f :X \to Y$ which is sequential continuous but not continuous where $X$ and $Y$ are some topological spaces.

Give an example of a function $f :X \to Y$ which is sequential continuous but not continuous where $X$ and $Y$ are some topological spaces. I have seen some example which uses $X$ to be non ...
3
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0answers
50 views

Computational Topology Codes

I am working on a project with a PI that thinks could be solved with computational topology tools. For this project, we will be looking at the persistent homology of objects in 3D images. I tried ...
2
votes
2answers
43 views

An example of open closed continuous image of $T_0$-space that is not $T_0$

Engelking in his "General Topology" states that $T_0$ separation axiom is not preserved under open closed continuous maps. But I can't find any example of open closed continuous image of $T_0$-space ...
1
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2answers
18 views

Cantor's Intersection Theorem

If the subsets of the compact space are already non-empty, isn't it obvious that the even the smallest subset is non-empty, and so the intersection is also non-empty because it would be the smallest ...
5
votes
2answers
43 views

Show that the collection of all open subsets of $X$ that are contained in $Y$ is a topology on $Y$.

This question is from a text book. Please let me know if my proof is vaild. Suppose $X$ is a topological space and $Y$ is an open subset of $X$. Show that the collection of all open subsets of ...
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0answers
35 views

Show that $y_n=x_{\phi(n)}$, defines a Cauchy sequence. [on hold]

Let $\phi:\mathbb{N}\to\mathbb{N}$, such that $\displaystyle\lim_{n\to\infty}{\phi(n)}=\infty$. If $(x_n)$, is a Cauchy sequence in the metric space $M$, then $y_n=x_{\phi(n)}$, defines a Cauchy ...
4
votes
1answer
138 views

Can a fractal be a manifold? if so: will its boundary (if exists) be strictly one dimension lower?

So it is weekend! and I am reading a nice book, "The Poincaré conjecture", written by a mathematician (Donal O'Shea, topologist). The book introduces step by step basic concepts of Topology, and talks ...
4
votes
2answers
90 views

Gluing diagrams: is it possible to glue a surface with itself in the same point? how is the diagram drawn?

I am learning the basic concepts of Topology, and playing now with the gluing diagrams (describing the fundamental domain of a topological space), this is an excerpt of a basic description I took from ...
2
votes
1answer
40 views

Looking for a clarification of the Suslin $\mathcal{A}$-Operation with a (finite) example

I have a problem concerning the output of (and the intuition behind) the Suslin $\mathcal{A}$-Operation. More specifically, I really don't see exactly what the output of it really is (even if I can ...
1
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0answers
48 views

Homotopic family of curves

I stumbled over the following question. Imagine we have a two homotopic curves on the sphere $\mathbb{S}^1$ namely $\gamma_1,\gamma_2$. Then we can write them as $\gamma_{i}(t) = e^{i \alpha_i (t)}$ ...
2
votes
2answers
33 views

How to prove that the subsets of $\mathbb{N}$ that don't contain arithmetic progressions of some length form closed sets of a topology?

I have exactly the same problem as this person, which I will rewrite below:Topology and Arithmetic Progressions. The reason I'm posting this is that I'm stuck at a later stage than the OP of that ...
1
vote
1answer
26 views

Is it true that factor spaces are T4 if product space is T4?

I use the following definition of $T_4$-space: for any two disjoint closed sets $A$, $B$ there exist disjoint open sets $U$, $V$ containing $A$ and $B$ respectively. Is it true that factor spaces are ...