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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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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22 views

Generalizing results about 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 ...
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40 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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19 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 < ...
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36 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, ...
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111 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 ...
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2answers
45 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 $\mathcal{R}=\{z\in\mathbb{C}:|p(z)|\leq R\}$, ...
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45 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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46 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, ...
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1answer
44 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 ...
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42 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 ...
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34 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 ...
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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 ...
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1answer
28 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 ...
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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 ...
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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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23 views

Convergence of sum of nets

In an arbitrary topological vector space $S$, if I have two nets ${x_\alpha}$ and ${y_\alpha}$ such that $x_{\alpha} \to x$ and $y_{\alpha} \to y$, can we say that $x_{\alpha} + y_{\alpha} \to x + ...
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2answers
73 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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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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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 ...
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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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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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89 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 ...
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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 ...
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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 ...
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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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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 ...
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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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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 ...
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48 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 ...
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2answers
32 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 ...
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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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45 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 ...
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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 ...
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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 ...
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1answer
43 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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20 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?
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3answers
44 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 ...
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115 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 ...
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Examples of generating the same topology

I'm teaching myself topology using a book I found. The question below is from the text. Then there are two additional questions that I am curious about. Please let me know if I'm doing it correct. ...
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Check a proof that, besides $\varnothing$, no open set in $\mathbb{R}^{n}$ has measure zero in $\mathbb{R}^{n}$

I am teaching myself Munkres's Analysis on Manifolds and came across an exercise, stated in the title of this question. Please see my proof below and, if doable, criticize it. That a set has measure ...
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1answer
46 views

subspace of a metric space

Let $(S,d)$ be a metric space, $\mathcal{S}$ the induced topology. $A\subset S$ a subset. It is easy to see that $A\cap\mathcal{S}=\mathcal{A}$, i.e., the topological subspace on $A$ is the ...
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1answer
36 views

How to determinate whether superset will be open or closed?

Let $M = (X, d)$ and A is closed subset of X, i.e. $A \subseteq X$. $A$ is told to be closed, iff it's complement $X\setminus A$ is open in $M$. But how can we determine, whether superset is open or ...
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53 views

Why is uncountable union of $\mathbb{R}$ the same as this space

Can anyone give an intuitive reasoning as to why the uncountable disjoint union of copies of $\mathbb{R}$ is the same as $\mathbb{R}$ with discrete topology product with $\mathbb{R}$ with the usual ...
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51 views

Axiomatic proof that all points of an open set are interior points

In "Principles of Mathematical Analysis, Rudin the following definition (f) to open sets: a set is open if all of its points are interior points Sidney Morris' Topology Without Tears, however, ...
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50 views

Proving that a set is open using epsilons.

I'm trying to prove that the set $$A=\{x=(x_{1},x_{2})\in\mathbb{R}^2:x_{1}^{2}+x_{2}^{2}>1\}$$ is open in $\mathbb{R}^2$ with the usual norm is open with the definition of "epsilons". My attempt ...
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1answer
14 views

Why is a convex subspace the requirement for equivalence beween subspace and order topologies?

I'm currently studying topology, and in one of the lectures we were presented with a theorem that went something like this (rephrasing since I don't have the theorem in front of me): Let $(X, ...