Understand Example 3.10 (b) Karl R. Stromberg, Chapter 3: Limits & Continuity

[Math Amateur]

I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...

I am focused on Chapter 3: Limits and Continuity ... ...

I need help in order to fully understand Example 3.10 (b) on page 95 ... ... Example 3.10 (b) reads as follows:

View attachment 9208

My question is as follows:Stromberg says that if $$X$$ is any set and $$\mathscr{T}$$ is the family of all subsets of $$X$$ ...

... then $$\mathscr{T}$$ is nothing but the metric topology obtained from the discrete metric ...Can someone demonstrate/explain exactly how/why this is true ...?

Help will be much appreciated ...

Peter

===================================================================================Example 3.10 (b) above refers to Example 3.2 (a) ... ... so I am providing access to the same ... as follows:

View attachment 9212
It may help readers of the above post to have access to Stromberg's definition of a topological space ... so I am providing access to the same ... as follows:
View attachment 9209

Stromberg's definition of a topological space refers to Theorem 3.6 ... ... so I am providing access to the statement of the same ... as follows:
View attachment 9210

Stromberg's definition of a topological space also refers to Definition 3.3 ... ... so I am providing access to the same ... as follows:

View attachment 9211

Hope that helps ...

Peter

 

[Opalg]

Stromberg says that if $$X$$ is any set and $$\mathscr{T}$$ is the family of all subsets of $$X$$ ...

... then $$\mathscr{T}$$ is nothing but the metric topology obtained from the discrete metric ...Can someone demonstrate/explain exactly how/why this is true ...?

Let $\rho$ be the discrete metric on the set $X$, and let $x\in X$. Then $\rho(x,y) = 1$ whenever $y\ne x$. Therefore $B_1(x) = \{y\in X:\rho(x,y)<1\} = \{x\}$. In words, that says that $B_1(x)$ is an open set consisting of the single point $x$.

Now let $S$ be a subset of $X$. Then $$S = \bigcup_{x\in S}\{x\} = \bigcup_{x\in S}B_1(x)$$. That is a union of open sets and is therefore open. Again putting it in words, this says that in the discrete metric every subset of $X$ is open.

So the topology obtained from the discrete metric (which by definition is the family of all open subsets of $X$) is the family of all subsets of $X$.

 

[Math Amateur]

Let $\rho$ be the discrete metric on the set $X$, and let $x\in X$. Then $\rho(x,y) = 1$ whenever $y\ne x$. Therefore $B_1(x) = \{y\in X:\rho(x,y)<1\} = \{x\}$. In words, that says that $B_1(x)$ is an open set consisting of the single point $x$.

Now let $S$ be a subset of $X$. Then $$S = \bigcup_{x\in S}\{x\} = \bigcup_{x\in S}B_1(x)$$. That is a union of open sets and is therefore open. Again putting it in words, this says that in the discrete metric every subset of $X$ is open.

So the topology obtained from the discrete metric (which by definition is the family of all open subsets of $X$) is the family of all subsets of $X$.

Thanks for a most helpful post, Opalg ...

Just reflecting on what you have said ...

Thanks again ...

Peter