Are the only sets to be considered open in a set X are the ones contained in the given topology?

Question

I'm new to topology, and I came across this definition on topology:

Given a topological space (X, T), a topology is defined to be a collection of open subsets which satisfies the following properties: 1. empty and X are in T 2. union of arbitrary collection of sets in T are open 3. intersection of a finite number of sets in T are open.

I'm quite confused if the only sets to be considered open are the ones in T, or some of the other subsets not in T can also be considered open. Can anyone give me light to this?

Also, are all sets in T are considered to be open and closed at the same time?

Thanks!

Answer 1

If $(X,\tau)$ is a topological space, then the open subsets of $X$ are the elements of $\tau$ and only the elements of $\tau$.

Answer 2

It is wrong and confusing if a topology $\tau$ on set $X$ is defined as a collection of open subsets of $X$ such that....

The word "open" must be left out since at time of defining you actually do not know yet what it means.

This misstep might have been the source of your confusion.

A topology $\tau$ on set $X$ should be defined as a collection of subsets of $X$ such that...

Afterwards the concept "open subset" can be defined. If set $X$ is equipped with a topology $\tau$ then a subset $A$ of $X$ is open if $A\in\tau$.