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
16 views

Confused with order topology

What does $0\times 1$ mean in the order topology $?$ How does ${{1}\over{2}} \times 0$ look like? Are they just a point or a line$?$ How do i visualize them$?$ I understand that $[0,1]\times[0,1]$ is ...
3
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2answers
41 views

If $\mathcal{B}$ is a base of a topology space $\left(X,\tau\right)$, then the Borel $\sigma$-algebra is generated by $\mathcal{B}$?

Let $\left(X,\tau\right)$ a topology space and $\mathcal{B}$ a base of the topology, my question is: The Borel $\sigma$-algebra is generated by $\mathcal{B}$ ?
2
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0answers
18 views

Find a set with empty interior and boundary equal to closure of $B^2$

I'm trying to find a set $A$ in $\mathbb{R}^2$ such that $\operatorname{Int}(A)$ is empty and $\operatorname{Fr}(A)=\operatorname{Cl}(B^2)$ I'm not sure how to do this. If I define my set as ...
2
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1answer
32 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 ...
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1answer
31 views

Convergence of filters in topological spaces [on hold]

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 ...
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0answers
78 views

If $\bar X$ is open, then $X=\bar X$. [on hold]

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 ...
3
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3answers
45 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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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
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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
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1answer
44 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. ...
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2answers
55 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 ...
3
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3answers
58 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
21 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
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2answers
41 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, ...
7
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1answer
149 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
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2answers
47 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 $\mathcal{R}=\{z\in\mathbb{C}:|p(z)|\leq R\}$, ...
3
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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 ...
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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, ...
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 ...
3
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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
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0answers
37 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 ...
2
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0answers
25 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 ...
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 ...
1
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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
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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 ...
1
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2answers
92 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 ...
2
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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 ...
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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 $E$. ...
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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
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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 ...
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1answer
20 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 ...
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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 ...
3
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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 ...
1
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2answers
19 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 ...
2
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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 ...
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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 ...
5
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2answers
44 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 ...
9
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1answer
64 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 ...
3
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0answers
51 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
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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 ...
2
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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 ...
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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 $\gamma_{i}(t) = e^{i \alpha_i (t)}$ ...
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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 ...
2
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1answer
42 views

Help me understand the reasoning used in the following lemma (38.1) from James Munkres' Topology.

Let $X$ be a space and $h: X \to Z$ be an embedding of $X$ in the compact Hausdorff space $Z$. There exists a corresponding compactification $Y$ of $X$ such that $H:Y \to Z$ is an embedding and equals ...
2
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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 ...
2
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1answer
45 views

2.25 of Lee's introduction to topological manifolds

If M is an n-dimensional manifold with boundary, then IntM is an open subset of M , which is itself an n-dimensional manifold without boundary. Here are the definitions to use: If M is an n-manifold ...
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0answers
21 views

C^1 mapping of a non-metric topological space - does this make sense?

Is there a way to define a derivative on a mapping between general topological spaces without invoking a metric?
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1answer
23 views

Continuity in general topological space (non-metric)

When defining continuity using open sets in a general topological space without a metric, is this considered C^0 or C^inf or something in between?
1
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3answers
46 views

Equivalence of norms problem.

How would I show that $\|\cdot\|_3$ and $\|\cdot\|_\infty$ are equivalent norms on $\mathbb R^2$? I understand that to say two norms are equivalent, then there exist two real constants, $m,M$ such ...
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 ...