Difference between revisions of "Isolated points"
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| − | Let $X \subset \mathbb{R}$. We say a point $x \in X$ is an isolated point if there exists a $\delta > 0$ such that $(t-\delta,t+\delta) \cap X = \emptyset$. It is known that for any such $X$, the set of isolated points of $X$ is [http://math.stackexchange.com/questions/402827/a-question-on-countability-of-isolated-points-of-a-subset-of-r at most countable]. | + | Let $X \subset \mathbb{R}$. We say a point $x \in X$ is an isolated point if there exists a $\delta > 0$ such that $(t-\delta,t+\delta) \cap X = \emptyset$. It is known that for any such $X$, the set of isolated points of $X$ is [http://math.stackexchange.com/questions/402827/a-question-on-countability-of-isolated-points-of-a-subset-of-r at most countable]. Note that all sets of isolated points are closd sets, for example $\left\{ \dfrac{1}{n} \colon n=1,2,\ldots \}$ is not closed, since $0$ is a limit points of this set. |
Revision as of 17:59, 20 May 2014
Let $X \subset \mathbb{R}$. We say a point $x \in X$ is an isolated point if there exists a $\delta > 0$ such that $(t-\delta,t+\delta) \cap X = \emptyset$. It is known that for any such $X$, the set of isolated points of $X$ is at most countable. Note that all sets of isolated points are closd sets, for example $\left\{ \dfrac{1}{n} \colon n=1,2,\ldots \}$ is not closed, since $0$ is a limit points of this set.
Let $\mathbb{T}=\{\ldots,t_{-1},t_0,t_1,\ldots\}$ be a time scale of isolated points with $t_k > t_n$ iff $k>n$. Define the bijection $\pi \colon \mathbb{T} \rightarrow \mathbb{Z}$, $\pi(t_k)=k$.
The set $h\mathbb{Z}=\{\ldots,-2h,-h,0,h,2h,\ldots\}$ of multiples of the integers is a time scale.
| Generic element $t\in \mathbb{T}$: | For some $n \in \mathbb{Z}, t=t_n$ |
| Jump operator: | $\sigma(t)=\sigma(t_n)=t_{n+1}$ |
| Graininess operator: | $\mu(t)=\mu(t_n)=t_{n+1}-t_n$ |
| $\Delta$-derivative: | $f^{\Delta}(t)=f^{\Delta}(t_n) = \dfrac{f(t_{n+1})-f(t_n)}{t_{n+1}-t_n}$ |
| $\Delta$-integral: | |
| Exponential function: |
