Questions tagged [general-topology]
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; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.
55,403
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$A \subset \mathbb{R}^n$ is bounded if and only if its closure $\overline{A}$ in $\mathbb{R}^n$ is compact.
I would like to prove that $A \subset \mathbb{R}^n$ is bounded if and only if its closure $\overline{A}$ in $\mathbb{R}^n$ is compact. This is problem 23.2 in Tu's book on manifolds. Here is my proof:
...
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Prove that (R, \tau cof ) is topological space
Advanced topology
Prove that (R, tau cof ) is topological space?
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16
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Relationships between completeness and closedness
Specifically. Suppose X is a metric space, A $\subseteq$ X .I want to show A is closed in X.
2
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2
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43
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Show that $A=\{(x,y)\in \mathbb{R}^{2}:y>x^2\}$ is open in $\mathbb{R}^{2}$ with the usual metric.
I want to prove that $A=\{(x,y)\in \mathbb{R}^{2}:y>x^2\}=\{(x,y)\in \mathbb{R}^{2}:y-x^2>0\}$. We then want to see that there exists $r>0$ such that $B((x_{0}, y_{0}); r) \subseteq \{(x,y)\...
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42
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Show that dist$(p,A)=0$ if and only if $p\in \overline{A}$
I want to proof that
$\mbox p \in \overline A \iff d(p, A)= \inf\{d(p,a):a\in A\}=0$
My attempt for the first implication is as follows. We know that $\overline A= A \cup A'$, it would suffice to ...
- 832
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1
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18
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Clarification of Proof about Continuous functions and Product Topology
So, I am trying to solve this problem, and I need some hint on how to prove $h^{-1}[U\times V] = f^{-1}(U) \cap g^{-1} (V)$ as shown in the first solution. I know I have to show mutual inclusion, but ...
- 55
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19
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Group of covering tansformations of universal covering for Riemannian manifold is total discontinuous?
Picture below is from Do Carmo's Riemannian Geometry. I use the definition of covering map. In this definition, if assuming $V_i\cap V_j =\emptyset ~(i\ne j)$ ($V_i$ is sheet), then I know how to get ...
- 5,487
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34
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Name for point in space which is contained in all nonempty open sets?
Suppose $(X, \mathfrak{T})$ is a topological space. Let $x \in X$ have the property that
$$\forall \, \, \mathrm{open} \, \, \varnothing \neq U \subseteq X, x \in U$$
In other words, $x \in \...
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Proof explanation: Every closed subset is a boundary
I'm reading the following proof here on Mathstack Exchange:
Let $Y$ be a topological space that can be partitioned into dense subsets $D$ and $E$. If $X \subset Y$ is closed, then there is a $V \...
- 2,201
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Proof that $\overline A= Int(A) \cup \partial (A)$
I want to prove that $\overline A= Int(A) \cup \partial (A)$, My attempt is as follows.
For the first contention. Be $x\in \overline A$, then for all $r>0$ $B(x; r) \cap A \neq \emptyset$ and let $...
- 832
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45
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Is $\{(z_1,z_2) \in \mathbb{C}^2 :|z_1|=|z_2| = 1, z_1^m = z_2^n\}$ a circle?
Fix $m,n \in \mathbb{Z}\setminus \{0\}$ and consider the subspace
$$
A = \{(z_1,z_2) \in \mathbb{C}^2 :|z_1|=|z_2| = 1, z_1^m = z_2^n\}
$$
of the $2$-torus. Is this space homeomorphic to a circle?
...
- 1,329
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20
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Relative topology on [0, 1]. Filters. Convergence.
Let $\mathscr F$ be the collection of subsets of $[0,1]$ which contain an interval $[\frac{1}{3},\frac{2}{3}]$. Use the relative topology on $[0,1]$ considered as a subspace of $\mathbb R$ with the ...
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32
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Torus inside $S^2\times S^2$
Suppose we have a torus $\mathbb{T}^2$ inside $S^2\times S^2$, i.e., there exists a continuous injective function $\phi:S^1\times S^1\rightarrow S^2\times S^2$. Is it always true that this torus can ...
- 4,456
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a bounded sequence that converge weakly [closed]
let E be a normed vector space such that E is isomorphe to E**, let $(x_n)_n$ a bounded sequence on E ,prove that we can construct a subsequence of $x_n$ that converge weakly.
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35
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Open sets. Hausdorff Space.
Consider the topology on $[-1, 1] \times [-1, 1] \subset \mathbb R^2$ where the set is considered open if and only if it is empty, or contains $(0,0)$.
Prove: This definition of open sets defines a ...
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3
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36
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smallest subcover to cover sphere using half spheres
Let $\mathbb{S}^n$ be the unit sphere in $\mathbb{R}^{n+1}$, and $\{C_{\alpha}\}_{\alpha\in A}$ be a covering of $\mathbb{S}^n$ consists of closed half spheres, i.e. $C_{\alpha}=\{v\in\mathbb{S}^n|v\...
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31
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Relation between basis and subbasis in topology
I have two questions about the relation of bases and subbases in topology:
1.) Every basis is also a subbasis (for the same topology), but how can I show that? It means that any finite intersection of ...
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29
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Metric space $l_p$ which is closed and bound and not compact?
Consider the space $l_p =\{(x_n): \sum_{n=1}^\infty \vert x_n\vert^p < \infty\}$ with the norm $\|.\|_p$.
Take: $A$ of all members in the sequence $(u_n)_n \subset l_p$ such that $u_n = (0,...,0,1,...
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24
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the additive of continuous maps [closed]
Let $X$ be a topological space. $f$ and $g$ are two continuous maps from $X$ to real number space $\mathbb R$. Prove that $f+g$ and $fg$ are also continuous maps from $X$ to $\mathbb R$.
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51
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In $\mathbb{R}$ with the usual metric, determine the closure of $A=(0, 1]$
I want to determine the closure of $A=(0, 1]$, I will consider the most appropriate definition of closure for this problem to be $Cl(A)=\text{Int(A)} \cup \partial(A)$. In this case, I consider that $\...
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30
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Characterization of Relative Compactness in Complete Metric Spaces
Let $(M, d)$ be a complete metric space and $A\subseteq M$. I want to show that $A$ is relatively compact (i.e., $\overline A$ is compact) iff there exists no infinite subset $S\subseteq A$ such that $...
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1
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32
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Some examples to demonstrate compact support
I am an electrical engineer trying to learn some real analysis from the internet. Please provide an instructive example of a function with compact support and another function without compact support. ...
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63
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$\mathrm{Top}$ as a double category
In notes of Susan Niefields about double categories (http://www.tac.mta.ca/tac/volumes/26/26/26-26.pdf) one of the first examples of a double category is a double category structure on $\mathrm{Top}$, ...
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105
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Nowhere dense subsets of a dense subspace are nowhere dense in the whole space and vice versa
It seems, with the following lemma, the proposition at the bottom easily follows.
If $Y\subset X$ dense. Then, for every nonempty $A\subset Y$, $\text{Int}_Y \left(\text{cl}_Y A\right)=Y\cap \text{...
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56
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Can it be determined whether the given function is necessarily a polynomial?
Consider a function $f\in C^\infty[0,1]$ satisfying the property that for every $x\in[0,1]$, there exists some $n\in\mathbb N$ such that $f^{(n)}(x)=0$. The question is whether $f$ must necessarily be ...
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1
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62
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Arbitrary union of measure $0$ open sets has measure $0$ in a separable metric space
Let $X$ be a separable metric space, with $(X, \tau)$ the corresponding topological space, and let $\mu$ be a (positive) measure on $(X, \mathcal{B}(X))$, where $\mathcal{B}(X)$ is the Borel $\sigma-$...
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14
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$Y$-Set problem from topology [duplicate]
For context, I am self studying some basic topology in preparation for taking Algebraic Topology (following Hatcher), and I've come across the following practice problem. The source is Math 327 (2019 ...
- 1,164
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25
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Upper bound to Smallest volume lattice polytope containing a hypersphere?
Let $S$ be a hypersphere of radius $r$ in $\mathbb{R}^n$ whose centre may or may not be a lattice point (i.e. a point in $\mathbb{Z}^n$). Can you give any upper bound to the volume of the smallest ...
- 2,304
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2
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35
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Comparing topologies and metrics
I have the space of continuous functions from $[0,1]$, $C([0,1])$, with 3 different metrics, $d_1$,$d_2$ and $d_\infty$, and the topologies induced by those metrics, $\tau_1$, $\tau_2$, $\tau_\infty$. ...
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22
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Topological conjugacy for vector field failing on the level of flows
Given two vector fields $F_1,F_{2}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with their flows $\phi_{1}, \phi_{2}:\mathbb{R} \times \mathbb{R}^{n}\rightarrow \mathbb{R}^n$
We say they are topological ...
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29
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About the limit in affine line $\mathbb{A}^1$.
When I am reading the answer Example of a non extendable rational map, there is a point which made me confused. That is $\lim_{t\to 0}tx=(\lim_{t\to 0})x$, where $t,x\in\mathbb{A}^1=\mathbb{K}$, under ...
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1
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42
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Is ∞ a limit point of $\mathbb R$ ? If not, how to understand Rudin's definition at the beggining of chapter $4$ (PMA)?
I am starting reading the $4^{th}$ chapter of PMA from Walter Rudin.
The chapter is about continuity and it defines $$\lim_{x\to a} f(x)$$ (for a function mapping a metric space $E$ into a metric ...
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1
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$\sigma$-finiteness of measures and separability of $L^p$ spaces
The fact that a measure $\mu$ is $\sigma$-finite determines or not the separability of $L^2(\mu)$? I proved that $L^2([0,1])$ endowed with the counting measure $m$ is not separable since it admits an ...
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1
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41
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Why does connected not imply locally connected?
The terms used is based on the definition given in Munkres’ Topology. In the book, he claims that the Topologist sine curve is connected but not locally connected. He opines that it is connected since ...
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38
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Equivalent definitions of Baire spaces
We say that a metric space $X$ is a Baire space if there is no open set $E$ such that $$E \subseteq \bigcup\limits_{n\geq 1} F_i,$$ in which each $F_i$ is a closed set with empty interior.
Suppose ...
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1
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Understanding Lemma 15.19 of the Gortz's Algebraic Geometry ( Second question : injectivity of a morphism under some condition )
I am reading the Gortz's Algebraic Geometry Lemma 15.19 and stuck at showing some reductibility of schemes to be reduced.
First of all, I propose somewhat basic question. Let $X$ be a topological ...
- 1,504
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28
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Image of a filter
Let $X=\prod_{i\in I} X_{i}$ be product topological space and $\mathcal{F}$ be a filter on $X$. As I read in some texts, the image Filter of $\mathcal{F}$ under any projection $\pi_{i}$ is the family ...
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1
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26
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Prove that $\tau = \mathcal{P}(Y) \cup \{X\}$ is a topology over $X$.
Let $Y$ be a infinite set, $x_0 \notin Y$, and define $X = Y \cup \{x_0\}$. Prove that $\tau = \mathcal{P}(Y) \cup \{X\}$ is a topology on $X$.
I don't see why the hypothesis "$Y$ infinite" ...
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13
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Open "enlargement" of locally finite family of compact sets
I want to show that
Let $M$ be a paracompact $T_2$ space and let $(K_i)_{i\in I}$ be a locally finite family of compact subsets of $M$, then there is a locally finite open covering $(U_j)_J$ of $M$ ...
3
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2
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123
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A closure of a set is closed?
Suppose the topological space $\left(X,\mathcal{T}_{X}\right)$ is given by $X= \left\{1,2,3,4,5\right\}$ and $\mathcal{T}_{X}=\left\{ \left\{ 1,2\right\} ,\left\{ 2,3\right\} ,Ø,X;\left\{ 1,2,3\right\}...
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35
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Understanding the meaning of a topology being weaker than another.
In topology, a topology $T_1$ is said to be weaker than another $T_2$ if $T_1$ is contained in $T_2$. The reason for "weaker" is that there could be more convergent sequences in $T_1$ than ...
- 2,878
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48
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Proving $\partial (A \cap B) \subset \partial(A) \cup \partial(B)$ using cases
Let $A$ and $B$ be two subsets of a topological space $S$. I would like to prove that
$$\partial (A \cap B) \subset \partial(A) \cup \partial(B)$$
where $\partial$ denotes the boundary of that set. I ...
- 2,797
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1
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25
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Prove existence of disjoint open sets containing disjoint closed sets
Let $(X, d)$ be a metric space, and let $F_1$ and $F_2$ be disjoint
closed subsets of X. Prove that there exist open subsets $U_1, U_2 ⊂ X$ so that
$F1 ⊂ U1, F2 ⊂ U2$ and $U1 ∩ U2 = ∅$.
Prove ...
- 2,149
1
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0
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35
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Anti-isometry that permits existence of metric space that inverts relations of another metric space
Suppose we have two spaces with respective metrics $d_1$ and $d_2$ and an (anti-)isometry $f$ that maps objects from the first space to the second. I'm interested in the setting where the following ...
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0
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19
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Relating Line and Circle Topologies
I'm trying to develop a better conceptual understanding for how a topology endows shape to a space. Contrasting discrete (too many open sets so points too far from each other), Hausdorff (just enough ...
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77
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Is there a name in English for this kind of compact set?
In French, they call “compact à bord régulier” a compact set $K \subset \mathbb{C}$ such that $K=cl(int(K))$ and $\partial K$ is a finite union of piecewise $C^2$, disjoint and orientable (relatively ...
- 1,340
-1
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0
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23
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Space of difference of continuous convex functions is dense in continuous functions
Let $X$ be a bounded closed convex subset of a locally convex space $E$. Let $\Gamma(X)$ be the set of continuous convex functionals on $X$ and let $C(X)$ be the set of continuous functionals.
Is $\...
- 4,683
-1
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1
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75
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Why universal covering space of $S^2$ is not $R^2$?
I know the universal covering space of $S^2$ is $S^2$, where $S^2$ is 2-dimensional sphere. But as picture below, I feel $R^2$ also be the universal covering space of $S^2$. But, obviously, $S^2$ is ...
- 5,487
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1
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190
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Is there such a theorem in general topology
I have proved that isolated point sets in a topological space $X$ (at least $T_1$) must be nowhere dense in $X$ (see this paper for more detail). The isolated point set of a set $A$ in $X$ is defined ...
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votes
1
answer
15
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In a TVS, the cylinder generated by a cross section of an open set is an open set
Because the direct sum of two topological vector spaces (TVS) equipped with the product topology is automatically a TVS,
I wonder whether the converse is true:
the topology of a TVS is the same as the ...
- 33