Definition of a Topology from Metric Spaces

Question

In Rudin's Real and Complex Analysis, he give a motivation for the definition of a topology using open sets in a metric space. He says, for a metric space $X$, and for a set $\tau$ of sets $E \subset X$ if

$(A)$ $\tau$ is the set of opens sets of $X$ (defined using neighborhoods and interior points)

then

$(B)$ $\tau$ is a topology. i.e.

$\,\,\,\,\,\,\,\,(i)$ $\,\,\,\,\emptyset \in \tau$, $X \in \tau$

$\,\,\,\,\,\,\,\,(ii)$ $\,\,\,\,\tau$ is closed under countable intersections

$\,\,\,\,\,\,\,\,(iii)$ $\,\,\,\,\tau$ is closed under countable and uncountable unions.

I understand the direction $(A)\implies(B)$. But is it necessarily true that if $\tau$ is a topology on a metric space $X$, then $\tau$ is the set of open sets. i.e. $(B)\implies(A)$. Or similarly, does $(B)\implies (A')$ where

$(A')$ $\tau$ is a set of open sets of $X$. (not the set of open sets).

If $(B)$ does not imply $(A)$ or $(A')$ then why do we use $(B)$ as the topological definition for open sets if it permits sets other than open in a metric space?

Answer 1

There are many ways of defining a topology on a set and, in particular, on a metric space. For instance, in $\mathbb R$, if $\tau$ consists of $\emptyset$, $\mathbb R$ and every interval of the form $(-\infty,a)$ or $(-\infty,a]$, then $\tau$ is another topology. Note that (in this example) some elements of $\tau$ are not open sets with respect to the usual topology on $\mathbb R$. Also, some open subsets with respect to the usual topology on $\mathbb R$ do not belong to $\tau$.

Answer 2

No. $B\not \implies A$.

The Euclidean metric is ONE possible topology on $R$ but it is not the only one. Any set of sets to satisfy B) will be a topology.

Consider the discrete topology where every set is both open and closed. $\tau =$ then set of all subsets. B) is certainly satisfied. $\emptyset \in \tau$ and $\mathbb R \in \tau$ and the intersection and union of any countable or uncountable union are intersection is a set.

Or consider the topology where $\tau = \{\emptyset, \mathbb R\}$ and $\emptyset$ and $\mathbb R$ are the only sets that are open or closed. B) is satisfied as $\emptyset\cap \emptyset = \emptyset, \emptyset \cup \emptyset = \emptyset, \emptyset \cap \mathbb R = \emptyset, \emptyset \cup \mathbb R = \mathbb R, \mathbb R \cap \mathbb R = \mathbb R; \mathbb R \cup \mathbb R = \mathbb R$.

Another not so trivial one is $\tau = \{\emptyset, \mathbb R\}\cup \{(-\infty, x)|x\in \mathbb R\}$.

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