Tag: topologia
-
If R is a local ring then R[[x]] is local.
Recall that if S is a commutative ring with unit, S is local if and only the set of non unit of S is an ideal, i.e, denoting with S* the unit of S, S-S* is an ideal of S. Now, let m be the maximal ideal of R. We prove that R[[x]] – R[[x]]*…
-
Is it true that any closed set of an arbitrary topological space X is the boundary of some subset of X?
In general this question has a negative answer: take a discrete topological space X and Z a non empty closed subset of X. Suppose Z=Fr(A) for some subset A of X, where Fr(A) denotes the boundary of A. Now since X is discrete, A is closed, which means Fr(A)⊂A but A is also open so…
-
Let K be an algebraically closed field and a,b n×n matrices over K. Then ab and ba have the same characteristics polynomial.
There are a lot of proof of this fact however here we want to gave a proof that use the machinery of Algebraic Geometry. Recall that we define the affine n space over K as Aⁿ=Kⁿ . This set can be endowed with a nice topology, namely the Zariski topology. This is the topology which…
-
For which n≥2 there is m∈ℕ such that SO(n)≅RPᵐ ?
In the following if M and N are differentiable manifold we write M≅N iff M and N are diffeomorphic. We know that SO(2)≅S¹≅RP¹ and SO(3)≅RP³ , are this the only possibility? Well, yes, a possible argument is the following: The universal covering of RPⁿ is Sⁿ, while that of SO(n) is Spin(n), which is an…
-
Six proofs that √2 is irrational
Dimostriamo che √2 è irrazionale usando 6 dimostrazioni diverse. 1° Proof: Suppose that √2 ∈ ℚ then there exist two coprime integers a and b such that √2=a/b. It follows that 2b²=a². So a² is even, that is, a is even, or a=2k with k being an integer. But then 2b²=4k², or b²=2k², from which…
-
Che forma ha il sottoinsieme dei punti di ℝ³ soddisfacenti l’equazione -Z⁴+8Z³+X²+Y²-16Z²=0 ?
Consideriamo il sottoinsieme S dei punti (X,Y,Z) di R³ soddisfacenti l’equazione -Z⁴+8Z³+X²+Y²-16Z²=0 . Che forma ha S ? Per capirlo prendiamo un punto (X,Y,Z) in S, allora: -Z⁴+ 8Z³ + X² + Y² -16Z² =0. Tale condizione diventa: X²+ Y² = Z⁴ -8Z³ + 16Z² = (Z² -4Z)² quindi posto v=Z, esiste u in [0,2pi)…
-
0 varietà topologiche
CLASSIFICAZIONE DELLE 0-VARIETÀ TOPOLOGICHE Ricordiamo che una varietà topologica di dimensione n∈ℕ (detta anche n-varietà topologica) è uno spazio topologico (M,τ) tale che sia: 1) N2 (cioè che ammetta una base numerabile ovvero che esista B⊆τ numerabile tale che ∀ V∈τ V è unione di elementi di B )2) T2 (cioè che ∀x,y ∈ M…
-
Horned sphere
Let A,B ⊆ IRⁿ be closed and homeomorphic as topological spaces equipped with the topology induced by the Euclidean one of IRⁿ. It is known that any homeomorphism h:A—>B extends to a homomorphism φ:IR²ⁿ—>IR²ⁿ. It follows that IR²ⁿ \ A is homeomorphic to IR²ⁿ \ φ(A) = IR²ⁿ \ h(A) = IR²ⁿ \ B .…
Marco Damele 