Proving Topology in X: A Look at Union & Intersection

In summary, the conversation discusses the definition of a topology in the set $X=\mathbb{R}\cup \{\star\}$ and how to show that it satisfies the properties of a topology. This includes showing that $X$ and $\emptyset$ are open, the union of two open sets is open, and the intersection of two open sets is open. The conversation also includes a correction to a statement regarding the union of intervals being a subspace of the real line.
  • #1
mathmari
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Hey! :giggle:

We consider the set $X=\mathbb{R}\cup \{\star\}$, i.e. $X$ consists of $\mathbb{R}$ and an additional point $\star$.

We say that $U\subset X$ is open if:

(a) For each point $x\in U\cap \mathbb{R}$ there exists an $\epsilon>0$ such that $(x-\epsilon, x+\epsilon)\subset U$.
(b) If $\star \in U$ then there is an $\epsilon>0$ such that $(-\epsilon , 0)\cup (0, \epsilon)\subset U$.

Show that this defines a topology in $X$.
We have to show that $X$ and $\emptyset$ are open, the union of two open sets is open and the intersection of two open sets is open.

First we show that $X$ is open :

(a) For each point $x\in X\cap \mathbb{R}=\mathbb{R}$ there exists an $\epsilon>0$ such that $(x-\epsilon, x+\epsilon)\subset X=\mathbb{R}\cup \{\star\}$. This is true since every neighboorhood of $x$ is contained.

(b) If $\star \in X$ then there is an $\epsilon>0$ such that $(-\epsilon , 0)\cup (0, \epsilon)\subset X=\mathbb{R}\cup \{\star\}$. Thisis true since the union of the intervals is a subspace of the real line.

Is that correct?

The emptyset is per definition open, or not? We don’t have to apply the given definition, do we? Let $M_1$ and $M_2$ be two open sets. We consider the union $M_1\cup M_2$. For each point $x\in M_1\cup M_2$ it is either $x\in M_1$ or $x\in M_2$ (or both) so statement (a) follows from the fact that $M_1$ and/or $M_2$ are open. The same holds also for statement (b).

Let $M_1$ and $M_2$ be two open sets. We consider the intersection $M_1\cap M_2$. For each point $x\in M_1\cap M_2$ it is $x\in M_1$ and $x\in M_2$ so statement (a) follows from the fact that $M_1$ and $M_2$ are open. The same holds also for statement (b). Therefore we get that the above defines a topology in $X$.

Is that correct and complete? :unsure:
 
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  • #2
No, you are completely missing the point! You are assuming that, because these are called "open sets", the properties of a toplogy must be true. This problem is asking you to prove that those properties hold for these particular sets. In particular, you have nowhere used the definitions of these sets.
 
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  • #3
mathmari said:
(b) If $\star \in X$ then there is an $\epsilon>0$ such that $(-\epsilon , 0)\cup (0, \epsilon)\subset X=\mathbb{R}\cup \{\star\}$. Thisi s true since the union of the intervals is a subspace of the real line.
Hey mathmari!

I think it is all correct except that this sentence can be improved. 🧐

$\star$ is given as an element of $X$, which is what we should state rather than say "if $\star\in X$". 🤔
And the union of the intervals is "subset" instead of a "subspace". 🤔
 

FAQ: Proving Topology in X: A Look at Union & Intersection

1. What is topology?

Topology is a branch of mathematics that studies the properties of geometric figures and spaces that are preserved under continuous transformations, such as stretching, bending, and twisting. It is concerned with the study of the shape and structure of objects, rather than their size or distance.

2. How is topology used in science?

Topology has a wide range of applications in various scientific fields, such as physics, engineering, biology, and computer science. It is used to model and understand complex systems, analyze data, and solve problems related to networks, surfaces, and shapes.

3. What is the union of topological spaces?

The union of two topological spaces is a new space that contains all the elements of both spaces. It is formed by taking the union of the two sets of points and the union of the two sets of open sets. The resulting space inherits its topology from the two original spaces.

4. What is the intersection of topological spaces?

The intersection of two topological spaces is a new space that contains only the elements that are common to both spaces. It is formed by taking the intersection of the two sets of points and the intersection of the two sets of open sets. The resulting space inherits its topology from the two original spaces.

5. How do you prove topology in X using union and intersection?

To prove topology in X using union and intersection, you need to show that the union and intersection of open sets in X satisfy the axioms of a topological space. This includes showing that the union and intersection of open sets are also open sets, and that the empty set and the entire space are open sets. Additionally, you need to show that the topology induced by the union and intersection of open sets is the same as the original topology on X.

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