Collection of open subsets of a set X contains X

Question

I'm studying functional analysis on my own and I'm going through Kreyszig's text "Introductory Functional Analysis with Applications".

$X$ is an arbitrary metric space.

In section 1.3 he states "It's not difficult to show that the collection of all open subsets of X, call it $\mathcal{T}$, has the following properties:

(T1) $\emptyset \in \mathcal{T}, X \in \mathcal{T}$

(T2) The union of any members of $\mathcal{T}$ is a member of $\mathcal{T}$.

(T3) The intersection of finitely many members of $\mathcal{T}$ is a member of $\mathcal{T}$."

I can't understand why it would be that $X \in \mathcal{T}$? Is this a typo? What if $X$ is a closed ball? Then it would seem $\mathcal{T}$ is an open ball and doesn't contain $X$.

Answer 1

The set $\mathcal T$ is the set of those subsets $A$ of $X$ such that$$(\forall a\in A)(\exists r>0):B_r(a)\subset A.$$Therefore, $X\in\mathcal T$; just take, for each $a\in X$, $r=1$. Then $B_r(a)\subset X$.

Answer 2

Definition: A topological space is a pair $A=(X,T)$ where $T$ is a collection of subsets of $X$ that satisfies $(T1), (T2), (T3).$ The members of $T$ are called the open sets of $A.$ For any $Y\subset X$ the topological space $B=(Y,T')=(Y, \{t\cap Y: t\in T\})$ is called a subspace of $A.$

The open sets of $B$ are not necessarily open sets of $A.$

I think a fairly common misunderstanding of this is due to the very common practice of referring to the space $(X,T)$ as "the space $X$" and calling $(Y,T')$ "the subspace $Y$".

It is also common to say "$Z$ is open in $X$" or "$Z$ is an open subset of $X$" instead of $Z\in T.$ And to say "$W$ is open in $Y$" for $W\in T'.$

A set is not intrinsically open or closed or neither.