Need help formalizing "T is an open set"

In summary, to show that T is an open set, you need to find an interval around P where every point is in T.
  • #1
Terrell
317
26

Homework Statement


Let ##S\subseteq \Bbb{R}## and ##T = \{ t\in \Bbb{R} : \exists s\in S, \vert t-s\vert \lt \epsilon\}## where ##\epsilon## is fixed. I need to show T is an open set.

Homework Equations


n/a

The Attempt at a Solution


Let ##x \in T##, then ##\exists \sigma \in S## such that ##x \in (\sigma -\epsilon, \sigma +\epsilon)##. Let ##\delta = \min(\frac{\vert \sigma -\epsilon -x\vert}{2}),\frac{\vert \sigma +\epsilon -x\vert}{2}## and consider the interval ##(x-\delta, x+\delta)##. If ##y \in (x-\delta, x+\delta)##, then
\begin{align}
\vert \sigma - y \vert \leq \vert \sigma -x\vert + \vert x- y\vert \lt \epsilon + \delta \cdots
\end{align}
I want to show ##\vert \sigma -y \vert \lt \epsilon##, but I am having trouble with finding the right expression to simplify this :cry:
 
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  • #2
Terrell said:

Homework Statement


Let ##S\subseteq \Bbb{R}## and ##T = \{ t\in \Bbb{R} : \exists s\in S, \vert t-s\vert \lt \epsilon\}## where ##\epsilon## is fixed. I need to show T is an open set.

Homework Equations


n/a

The Attempt at a Solution


Let ##x \in T##, then ##\exists \sigma \in S## such that ##x \in (\sigma -\epsilon, \sigma +\epsilon)##. Let ##\delta = \min(\frac{\vert \sigma -\epsilon -x\vert}{2}),\frac{\vert \sigma +\epsilon -x\vert}{2}## and consider the interval ##(x-\delta, x+\delta)##. If ##y \in (x-\delta, x+\delta)##, then
\begin{align}
\vert \sigma - y \vert \leq \vert \sigma -x\vert + \vert x- y\vert \lt \epsilon + \delta \cdots
\end{align}
I want to show ##\vert \sigma -y \vert \lt \epsilon##, but I am having trouble with finding the right expression to simplify this :cry:
IMO you're using too many symbols: x, y, ##\sigma, \epsilon, \delta##. Let's say you have an interval T of radius r, centered at P: (P - r, P + r). For any point x in this interval, there is some ##\epsilon## so that every point in the interval ##(x - \epsilon, x + \epsilon)## is also in the interval T. The same would not be true of closed intervals.
 
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  • #3
Mark44 said:
For any point x in this interval, there is some ϵϵ\epsilon so that every point in the interval (x−ϵ,x+ϵ)(x−ϵ,x+ϵ)(x - \epsilon, x + \epsilon) is also in the interval T.
Am I trying to do too much by using the triangle inequality?
 

Related to Need help formalizing "T is an open set"

1. What does it mean for a set to be "open" in mathematics?

In mathematics, an open set is a set that does not contain its boundary points. This means that for any point in an open set, there exists a small enough neighborhood around that point that is entirely contained within the set. In other words, there are no points on the edge or boundary of the set.

2. How is the concept of "openness" related to topology?

The concept of open sets is fundamental in the field of topology. Topology is a branch of mathematics that studies the properties of spaces that are preserved under continuous transformations, such as bending, stretching, and twisting. Open sets are used to define topological spaces and to study their properties.

3. What is the notation used to represent an open set?

In general, the notation used to represent an open set is T. However, this can vary depending on the context and notation conventions used in a particular mathematical field. Some common notations for open sets include O, U, and Ω.

4. How do you formally define an open set?

An open set can be formally defined as a set S such that for every point x in S, there exists a positive real number r (denoted as r > 0) such that the entire open ball centered at x with radius r is contained within S. In other words, every point in an open set has a small enough neighborhood that is entirely contained within the set.

5. Why is the concept of open sets important in mathematics?

The concept of open sets is important in mathematics because it allows for the rigorous study of continuity and convergence of functions. The openness of a set is closely related to the continuity of a function, and open sets are used to define important mathematical concepts such as limits, derivatives, and integrals. Additionally, open sets play a crucial role in topology and the study of geometric and topological properties of spaces.

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