Recent content by jamilmalik

So by construction, ##\rho(x,y) = d(x,y)## if both ##x,y \in C## or ##x,y \in X \setminus C##. As for considering the cases where ##x,y,z## are not all in the same component, my book defines component of a topological space as a connected subset ##C## of ##X## which is not a proper subset of...

Yes, we are using topological equivalence. To be honest, I do not think I have seen strong equivalence before. Thank you for this clarification.

Ok, so if ##C \cup X \setminus C = X##, then this creates a separation, according to what I am reading from Fred H. Croom's Principles of Topology. How do I tie this together with metric spaces to show equivalence? Again, many thanks for your feedback.

Would it have something to do with being connected? My textbook states that a space ##X## is connected if there do not exist open subsets ##A, B## of ##X## such that ##A \neq \emptyset, \quad B \neq \emptyset, \quad A \cap B = \emptyset, \quad A \cup B = X##. Is this equivalent? Thank you for...

Homework Statement Prove that if ##(X,d)## is a metric space and ##C## and ##X \setminus C## are nonempty clopen sets, then there is an equivalent metric ##\rho## on ##X## such that ##\forall a \in C, \quad \forall b \in X \setminus C, \quad \rho(a,b) \geq 1##. I know the term "clopen" is not a...

Hello everyone, I was wondering if I could get a simple introduction to this Theorem since I will have to be giving a presentation on it within the next month. Based on the statement itself, there is an assumption made in the hypothesis which is something I haven't quite understood yet: If...

Yes, I think it would be equal to ##E##. However I am skeptical if it was really that simple. I don't think I actually proved anything, but rather assigned a variable for each collection of unions.

What do you mean by an intersection was intended there?

Thanks for the responses. As noted above, I should add some more information regarding the terminology being used. A box in ##\mathbb{R}^d## is a Cartesian product ##B:=I_i \times \ldots \times I_d## of ##d## intervals ##I_1, \ldots , I_d##. The volume ##|B|## of such a box ##B## is defined as...

Homework Statement I am struggling with what seems like a very simple problem from Terrence Tao's Introduction to Measure Theory book (which is available for free online by the way). What I am trying to prove is the following: Give an alternate proof of Lemma 1.1.2(ii) by showing that any two...

Find the Laurent series of the following function in a neighborhood of the singularly indicated, and use it to classify the singularity. Homework Statement f(z) = \frac{1}{z^2-4} ; z_0=2 Homework Equations Laurent series \sum_{-\infty}^{\infty} a_n (z-c)^n The Attempt at a Solution I...