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by Nikos Athanasiou
Thu Nov 26, 2015 12:49 am
Forum: General Topology
Topic: Homeomorphism
Replies: 2
Views: 905

Re: Homeomorphism

Since the homeomorphism relation is an equivalence relation, suffices to show both spaces are homeomorphic to the open interval $(0,1)$. The latter space certainly is. For the first one, wlog $\phi = 0$ and parametrise $X$ as $ \lbrace (\cos x , \sin x) \mid 0<x< 2 \pi \rbrace $ . Define $f : X \rig...
by Nikos Athanasiou
Thu Nov 26, 2015 12:38 am
Forum: General Topology
Topic: Discrete metric space
Replies: 1
Views: 688

Re: Discrete metric space

In a discrete metric space, all points are open sets. Take the open cover $$ \lbrace \lbrace x \rbrace \mid x \in X \rbrace$$.

This does not have a finite subcover, since $X$ contains infinitely many points.
by Nikos Athanasiou
Thu Nov 26, 2015 12:30 am
Forum: Multivariate Calculus
Topic: Rellich-Kondrachov compactness theorem
Replies: 0
Views: 582

Rellich-Kondrachov compactness theorem

Let $\mathcal{U} \subset \mathbb{R}^d$ be an open bounded domain. Prove that $H^1(\mathcal{U}) \subset \subset L^2(\mathcal{U})$ ,i.e. the inclusion is compact, where $H^1$ is the Sobolev space $W^{1,2} $ and $L^2$ is our usual space on the corresponding domain. In other words, show that every unifo...
by Nikos Athanasiou
Thu Nov 26, 2015 12:17 am
Forum: General Topology
Topic: Metric topology
Replies: 2
Views: 960

Re: Metric topology

One direction is easy. If $f=0 $ on $D$ , then by continuity $f = 0$ on $\bar{D} = X$ Now assume the converse. A different definition of density in a metric space is the following : "$D$ is dense iff every open set in $X$ intersects $D$ non-trivially". So assume $D$ is not dense and pick an open set...
by Nikos Athanasiou
Tue Nov 24, 2015 5:34 pm
Forum: Number theory
Topic: Solve the equation
Replies: 0
Views: 801

Solve the equation

(i) Let $K = \mathbb{Q} (a)$ , where $a$ is a root of $X^2 - X + 12$ in $\mathbb{R}$ . Compute the class group of K.

(ii) Show that the equation $y^2 +y + 12 =3 x^5$ does not have any integer solutions.
by Nikos Athanasiou
Tue Nov 24, 2015 5:30 pm
Forum: Number theory
Topic: Find an integral basis
Replies: 0
Views: 660

Find an integral basis

Hello everyone! This is my first post. I salute the initiative of creating a forum solely dedicated to university mathematics. I feel it is something that has been missing for a long time. That being said, here's a nice exercise in number fields : Let $K = \mathbb{Q} (\sqrt{p}, \sqrt{q}) $ where $p ...