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8:34 AM
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In the mathematical field of point-set topology, a continuum (plural: "continua") is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space. Continuum theory is the branch of topology devoted to the study of continua. == Definitions == A continuum that contains more than one point is called nondegenerate. A subset A of a continuum X such that A itself is a continuum is called a subcontinuum of X. A space homeomorphic to a subcontinuum of the Euclidean plane R2 is called a planar continuum. A continuum X is homogeneous if for every two points x and y...
Q: A Local Connectedness Condition for Compact, Connected Metric Spaces

John SamplesI am having trouble proving a result from a paper, which of course includes no proof. I wonder if the author had a simple - but flawed - argument in mind, or if I'm just being a dunce. It is Theorem 3 here: https://www.jstor.org/stable/2372339 It involves the following property: If $X$ is a m...

Q: Conditions Weaker than Locally Euclidean

John SamplesThis is a very general question, but hopefully some people find it interesting. I'm working in the setting of compact metric spaces, so most of the basic topological properties will be satisfied. When the space $X$ is also connected, then I know that being locally connected is sufficient to be ...

Q: Local Bases in Locally Connected and Semi-Locally Connected Spaces

John SamplesI was trying to come up with a counterexample for something and kept failing and failing, so I was hoping someone could help me out. The setting is compact, connected, metric spaces (continua) so everything is nice. Here are the definitions I'm using: $X$ is locally connected at $x \in X$ if i...

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Q: Distributivity of intersections on unions of open scheme

PropositionXI aim to prove $P ∩\bigcup_α Q_α =\bigcup_α (P ∩Q_α)$. For $P,Q$ to be open subschemes of an affine scheme $X$. I have been told that every open scheme is a union of complements of hypersurfaces, an I have attempt write all the $Q_\alpha$'s to be a union of complement $S$ of some hypersurface, ...

Q: inverse laplace transformation of this expression

Urgent$X_2(s)=-\sum_{i=0}^{h-1}N^is^i(N+U(s))$, where $N$ is nilpotent matrix of order $h$, Could anyone help me to understand inverse laplace transformation of $X_2(s)$? $\delta$ is dirac delta function. Thanks for helping. I am attaching the actual expression. But I guess I will understand the origi...

Q: Are Brownian motion correlations preserved under a measure change?

Lost1Let $W$ and $B$ be two Brownian motions with $\text{d}\langle W, B\rangle_t = \rho \text{d}t$ under some probability measure $\mathbb{P}$, where $\rho$ is a constant. Let $\mathbb{Q}$ be an equivalent measure to $\mathbb{P}$. Does $\text{d}\langle W, B\rangle_t = \rho \text{d}t$ under $\mathbb...


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