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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2
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1answer
19 views

Is any subspace of a weak Hausdorff space necessarily weak Hausdorff?

As the title suggests, is any subspace of a weak Hausdorff space necessarily weak Hausdorff. Thanks.
3
votes
0answers
16 views

Passage to fixed point spaces is object function of a contravariant functor?

Let $X$ be a $G$-space. What is the easiest way to see that that passage to fixed point spaces, $G/H \mapsto X^H$, is the object function of a contravaraint functor $X^{(-)}: \mathscr{O}(G) \to ...
4
votes
1answer
48 views

Sequence of topological spaces

A friend of mine did an exercise where a part of the text was: In $\mathbb{R}^3$, with euclidian topology, we consider $X=\mathbb{S}^2 \setminus \{ N \}$, where $N= (0,0,1)$ and $E=\{(x,y,z) \in ...
-1
votes
3answers
91 views

Is $\mathbb R$ with usual topology a Hausdorff space?

$\mathbb R$ with usual topology is a Hausdorff space. By the definition of Hausdorff space, for any two distinct point in $\mathbb R$ we can find disjoint neighbourhoods... Now consider the set ...
-4
votes
3answers
57 views

How to define a continuous map from $[0,1] $ onto/into $\mathbb R$? [on hold]

It is possible to define a continuous map from $[0,1]$ onto $\mathbb R$?
1
vote
2answers
38 views

Manifolds with a finite but not trivial fundamental group

I came across this nice result: Theorem: If $M$ is a connected smooth manifold with finite fundamental group, then its first de Rham cohomology is trivial: $$H^1_{dR}(M)=0.$$ However, I don't ...
0
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0answers
21 views

Topology generated by basis equals intersection of all topologies that contain A

Here is my proof I was wondering any critiques to my proof. If A is a basis;The topology generated by A equals collections of all unions of elements of A that is $\tau = \bigcup_{i \in I: B_i \in ...
2
votes
1answer
43 views

Why is the measure of a boundary of an open ball positive in only a countable number of cases?

Let $X$ be a Polish (complete separable metric) space and $\mathbb{P}$ a Borel probability measure on $X$. Let $x_1, x_2, \ldots$ be a sequence of points dense in $X$. How can you prove that there is ...
1
vote
1answer
51 views

Is every simply connected open subset of $\Bbb R^n$ contractible?

Question: Is every simply connected open subset of $\Bbb R^n$ contractible? I know the result is true for $\Bbb R^2$ because by the Riemann Mapping Theorem every simply-connected proper open ...
3
votes
0answers
19 views

$\mathscr{O}(G/H, G/K) \cong (G/K)^H?$

What I am about to ask is related to the question presented here. Let $X$ be a $G$-space, where $G$ is a (discrete) group. For a subgroup $H$ of $G$, define$$X^H = \{x : hx = x \text{ for all }h ...
-4
votes
0answers
18 views

continuity of the piecewise functions [on hold]

$1$. $g(x)=0$,if $x$ is irrational and $g(x)=x$ if $x$ is rational Find all points of at which $f$ is continuous. $2$. Let $A$ and $B$ be compact sets. Define $A+B =$ ...
1
vote
1answer
24 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 ...
4
votes
2answers
51 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
votes
0answers
19 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 ...
3
votes
1answer
33 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 ...
-1
votes
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 ...
-2
votes
0answers
81 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
votes
3answers
46 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 ...
-1
votes
0answers
35 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
46 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. ...
1
vote
2answers
64 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. 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
votes
3answers
64 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 ...
1
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0answers
23 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
42 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
votes
1answer
155 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
49 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
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 ...
1
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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
votes
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
votes
2answers
48 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
0answers
39 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
votes
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
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 ...
1
vote
2answers
98 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
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 ...
0
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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 $E$. ...
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
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3answers
35 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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votes
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 ...
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 ...
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 ...
1
vote
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
votes
0answers
51 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
votes
2answers
45 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
votes
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
votes
0answers
56 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 ...