All Questions
Tagged with gt.geometric-topology at.algebraic-topology
1,145 questions
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Ways to prove that $n$-component Brunnian link is nontrivial
The attached image shows a way to construct an $n$-component Brunnian link for any $n\geq 3$. That is, this link is not trivial, but deleting any of its components makes the new link trivial. The ...
- 214
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2
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334
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Examples of finite polyhedra with finitely generated simple fundamental group
For $n\geq 2$, $P\mathbb{R}^n$ is a simple example of finite polyhedron with finitely generated simple fundamental group. I was wondering if someone could give me an example of a finite polyhedron ...
- 1,182
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1
answer
190
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Approximations by compact sub-spaces
Suppose $X$ is a Hausdorff (I'm happy to also assume "non compact") topological space that can be written as the topological direct limit
$$\varinjlim_{a\in J} K_a$$
for $J$ a directed set ...
user335418
1
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2
answers
262
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Reference request for widely used theorem
I am looking for a reference to the theorem that any oriented closed surface of genus $g$ is a 2-fold cover of $S^2$ (branched over 2$g$+2 points).
- 31
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1
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255
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Questions about a few terminologies in "Knots and Links" by Rolfsen
In "Knots and Links" by Rolfsen, he mentioned words like *"the collar of a boundary", "bicollared boundary", "a bicollar on the boundary". I just wonder what ...
- 15
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1
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326
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Addition of two homology classes is zero in construction of Poincare Sphere
I ask here the question since it hasn't been answered in
Math Stack Exchange.
I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one ...
- 909
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1
answer
2k
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Homology and homotopy type for knot complements
I'm trying to understand a paper by Kawauchi and Matumoto, 'An estimate of infinite cyclic coverings and knot theory.' In one of their proofs they have a manifold $E$ which is the complement of a ...
- 1,025
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1
answer
256
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N_3 and N_4 periodic and pseudo Anosov auto-homeomorphisms
It is well know that the genus three non orientable surface, N3, has only periodic and reducible auto-homeomorphisms, meanwhile the surface N4 is the first non orientable surface with pseudo Anosov ...
- 345
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1
answer
56
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Compatibility of two cylindrical regions
Let $M^2,N^2$ be connected closed surfaces. Suppose there exists region $D$ in the interior of $M \times [-2,2]$ such that (a) $D$ is homeomorphic to $N \times [0,1]$; (b) $D$ contains $M \times [-1,1]...
- 891
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102
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Approximate Jordan-Brouwer theorem (corrected)
My first attempt to ask this question sort of failed (I'll explain below).
This came up when thinking about this question.
It is well-known that the image of a homeomorphic embedding $f:S^n\to \mathbb{...
- 5,529
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1
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240
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free group actions on a contractible topological space [closed]
Let $\Sigma_k$ be the symmetric group on $k$-letters. Let $W$ be a contractible topological space with a free $\Sigma_k$-action (from the left). Let $X$ be a $CW$-complex and let $X^k$ be the ...
- 1,990
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1
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655
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Topological razors (ball-like spaces)
Introduction
Many admire the Euclidean space, and I am not an exception. I will try to catch the topological roundness of the $n$-ball in its greatest generality. I call the resulting axiomatized ...
- 7,500
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1
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595
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When is a bijective map between bundles a homeomorphism?
Let $F \rightarrow E_i \rightarrow X_i$ be a bundle with fibre $F$ for i=1,2.
Let $f:E_1 \rightarrow E_2$ be a bijective continuous map and $h: X_1 \rightarrow X_2$ a homeomorphism.
Is f then also ...
- 165
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1
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304
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good perspective in viewing manifolds of infinite dimension
Borel conjectued aspherical closed manifolds are topologically rigid.(i.e.a homotopy equivalence between two aspherical manifolds is homotopic to a homeomorphism).
now,soppuse M is a K(G,1) space,
it ...
- 179
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1
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414
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Equivariant maps inducing isomorphism in integral cohomology
Consider the following statement.
Suppose $X$, $Y$ are finite CW-complexes with free involution
and $\mu:X\to Y$ is an equivariant map.
If $\mu^*:H^i(Y;\mathbb{Z})\to H^i(X;\mathbb{Z})$ is an ...
- 11
1
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0
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61
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Map from simplex to itself that preserves sub-simplices: revisited
Here it is proved that, if $f$ is a continuous map from an $n$-simplex $\Delta$ to itself, that maps each sub-simplex of $\Delta$ to itself, then $f$ must be onto $\Delta$ (surjective).
I would like ...
- 3,549
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48
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Connected pre-images spanning $n$-cubes under dimension reducing maps
Let $I^n = [0,1]^n$ be the $n$-dimensional hypercube. For a continuous function $f: I^n \to \mathbb{R}^m$ with $m < n$, we're interested in the existence of points $p \in \mathbb{R}^m$ whose ...
user530766
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0
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57
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extendability of fibre bundles on manifolds with same dimensions
Let $M$ be an $m$-manifold. Let $M'\subseteq M$, where
$M'$ is also an $m$-manifold.
Let $N$ be an $n$-manifold. Let $N'\subseteq N$, where
$N'$ is also an $n$-manifold.
Suppose there is fibre ...
- 123
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0
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132
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The equation of cubic surface
I was studying the nature of singularities of cubic hypersurface on $\Bbb P^{n+1}$. The chosen form was
$$f(x_0,x_1 \dots , x_{n+1})=x_0^3+x_1^3+\dotsb+x_{n+1}^3-c(x_0+x_1+\dotsb+x_{n+1} )^3=0.$$
I ...
- 21
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61
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Necessary or sufficient conditions for the $k$-fold intersection to be empty in a covering with a "tree structure"
Consider a finite collection of $d$-dimensional balls $\mathfrak{B}=\{B_1,\ldots,B_n\}$ which cover a PL $d$-manifold $M$, i.e. $M=\bigcup_{i=1}^{n}B_i$. Suppose we want to compute the Euler ...
- 159
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145
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Determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?
What and how to determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?
Namely, how do we know
$$
K(Z_2,1)?, \quad K(Z_2,2)?, \quad K(Z,4)?
$$
Naively -- in each step ...
- 447
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0
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160
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Higher dimensional Seifert surfaces and link numbers of higher knots
In 3-manifold topology, the notion of Seifert surface is well known. It is then used to define link numbers of knots.
Question: Consider embedding $N^n \rightarrow M^{2n+1}$ of n-dimensional manifold $...
- 593
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97
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about codimension two foliation
Are there examples of codimension 2 foliations on closed compact 4-manifolds or 5-manifold
I am curious about examples of codimension
Are there any previous studies or lecture notes of foliation ...
- 73
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0
answers
78
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Diffeomorphism induced by small perturbation
Consider the surface $S_{\epsilon}$ defined as:
\begin{align}
%S&=\{\vec x \in \mathbf{R}^3: x=0\}, \\
S_{\epsilon}&=\{\vec x \in \mathbf{R}^3:\epsilon (x^2 + y^2 + z^2 - 1) + x=0\}.
\end{...
- 521
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0
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157
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Decomposing the homology of a connected sum of surfaces in a way which highlights the combinatorics of gluing
For each $i = 1,2$ and $j = 1,\ldots,n$, let $C_{i,j}$ be a connected compact oriented surface with $k$ boundary components. For $i = 1,2$, let $C_i = \sqcup_{j=1}^n C_{i,j}$, and let $C$ be the ...
- 7,527
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284
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A question on existence of gradient vector field on manifold with boundary
Let $M$ be a compact manifold with smooth boundary $\partial M$. Does $M$ admit a gradient vector field $\nabla u$, which has no zeros, i.e. $\nabla u(x)\neq 0$, $\forall x\in M\cup\partial M$?
Thanks ...
- 51
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143
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End space of non-compact 2-manifolds described with proper rays
I am wondering if the classification of noncompact surfaces given by Ian Richards can be stated with proper rays instead of nested sequences of connected open subsets with compact boundary. I asked ...
1
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0
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102
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DA structure of a Dehn twist
I am trying to find the DA bordered homology structure of a Dehn twist.In https://arxiv.org/pdf/0810.0687v6.pdf page 255 bottom the authors tabulate the differentials of the right module(A side) of ...
- 99
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97
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Homotopy type of complement to a union of linear subspaces
Im not sure if this question is appropriate for MO, but I'm looking for a hint about some questions about homotopy type of complement to a union of linear subspaces in vector space $\mathbb{R}^n, \...
- 11
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160
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Contractible four-manifold which admits a decomposition
Let $M^4$ be a noncompact, contractible, smooth manifold. Suppose there exists an exhaustion $M=\bigcup_{i\ge 1} U_i$ by open sets such that (1) $\bar U_i \subset U_{i+1}$ and (2) each $U_i$ is ...
- 891
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129
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Open cone homeomorphic to the Euclidean space
Let $X$ be a topological space and the open cone $C(X)$ over $X$ is defined to be $X \times [0,1)$ with $X \times \{0\}$ identified. Suppose $C(X)$ is homeomorphic to $\mathbb R^4$, can we prove that $...
- 2,535
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297
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Boundary map in Mayer-Vietoris sequence of cohomology
Suppose $M$ is a 3-manifold with connected boundary. Let $T$ be a tangle in $M$, i.e., $T$ is a embedded connected 1-submanifold whose boundary is on $\partial M$ ($T$ is not closed). Moreover, ...
- 673
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151
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Density of $G$-invariant morse functions
Let $G$ be a finite group acting on a compact manifold $M$. Let $f$ be a $G$-invariant smooth function. Can it be approximated by $G$-invariant Morse functions?
- 99
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80
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A characterization for a space that is similar to locally connected spaces
Let $X$ be a $T_0$ topological space with the property that there exists a basis $\{O_i\}_{i\in I}$ for $X$ such that for each $J\subseteq I$ the subspace $\bigcap_{i\in J}O_i$ has only finitely many ...
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152
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Complement of contractible locally Euclidean subspace
Let $X$ be a connected closed topological manifold. Let $S\subset X$ be a contractible locally Euclidean subspace. Is $X\setminus S$ connected?
- 11
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154
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Homotopy groups of ball complement
Let $X$ be a connected closed topological manifold. Let $n$ be an integer such that $\pi_i(X)=\{0\}$ for $1\leq i \leq n$.
Let $f:B^m\to X$ be a topological embedding, where $B^m$ is the $m$-...
- 19
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137
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Covers of a 4-manifold pull back a cohomology class to any algebraic multiple
Fix an algebraic integer $x\neq 0$. Is there a closed smooth 4-manifold $M$ with a class $\rho\in H^{1}_{\mathrm {dR} }(M)$ and a smooth covering map $\phi:M\to M$ such that $\phi^*\rho=x\rho$?
Is ...
user164740
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0
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53
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Spaces that are comparable with their compacts
This is an outgrowth of this question.
For a (metrizable) space $X$ consider the following increasingly strong properties:
(i) For every compact $K\subset X$ there is a map $f:X\to X$ such that $K\...
- 5,529
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143
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A topological property of curves on the plane $\mathbb{R}^2$
Let $\gamma\colon [0,1]\to \mathbb{R}^2$ be a continuous injective map.
Is it true that for any inner point $t\in (0,1)$ there exist an open neighborhood $U$ of $\gamma(t)$ and a homeomorphism $f\...
- 21.8k
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73
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Cyclic homotopies of quotients of $S^3$
We are given a free action of an abelian finite group on $S^3$. Let $L$ denote the quotient space and let an element $\alpha \in \pi_1 L =G$ be given. Does there exist a cyclic homotopy $h_t:L \to L$ ...
- 268
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123
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Action of the symmetric group on connected sums of manifolds (minus a disk)
Let $M$ be a connected compact topological $n$-dimensional manifold without a boundary and with a CW-structure $M= \bigcup M^i$. We have that
$$ (\#^g M)\smallsetminus D^n \simeq \bigvee_{i=1}^gM^{n-...
- 523
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59
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Integer valued signature of $4n$ dimensional orbifolds
Let $M^{4n}$ be a smooth oriented $4n$-dimensional manifold without boundary. Then we have an intersection form in $H^{2n}(M^{4n},\mathbb R)$ and such a form has signature $(n_+, n_-)$.
Question. I ...
- 14.3k
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105
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The inverse image of a Noetherian topological space
A topological space $X$ is called Noetherian if
closed subsets satisfy the descending chain condition, equivalently,
the open subsets satisfy the ascending chain
condition.
Let $A$ and $B$ be ...
- 11
1
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0
answers
152
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Stone cech compactification of a zero dimensional topological space
Let $X $ be a zero dimensional topological space, that is, a topological space with a basis of clopen sets. Is there any characterization for the ston cech compactification for such a space?
- 11
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89
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On the Multi-Compression theorem of Rourke and Sanderson
Suppose $M$ is a compact manifold so that it admits an embedding $f:M\to N\times\mathbb{R}^l$ with a splitting of the normal bundle as $\nu_f\simeq\nu\oplus\epsilon^l$ where $\nu$ is some $k$-...
- 3,173
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177
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Why is any one Wirtinger relation a consequence of the remainder? [closed]
As the question title suggests, how do I see that any one Wirtinger relation is a consequence of the remainder?
- 19
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0
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101
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coefficient of homology of configuration spaces over real projective spaces
In the slides Characteristic Classes of Surface Bundles
and Configuration Spaces, Miguel A. Xicot'encatl, page 38, what is the coefficient of the following homology?
Could the coefficient be an ...
- 1,990
1
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0
answers
126
views
cohomology ring of compact submanifolds of Euclidean spaces
Suppose we have a compact $m$-dimensional submanifold $M$ of $\mathbb{R}^N$ and we want to know the cohomology ring $H^*(M;\mathbb{Z})$.
Let $\epsilon>0$ and a $m$-dimensional finite simplicial ...
- 1,990
1
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0
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278
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Homology of spherical braid groups
By the spherical braid group, I mean the fundamental group of the configuration space of distinct unordered points in $S^2$. I am wondering what is known about the group homology of the spherical ...
- 116
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0
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114
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When Max(R) is Hausdorff space? [duplicate]
Let $R$ be reduce commutative ring with identity (a commutative ring such that $a^n$=0 ($a\in R$) implise $a=0$) and $Max(R)$ be the set of all maximal ideals of $R$. The hull-kernel (or Zariski ...
- 11