Finding counterexamples by definitions

Question

I'm a little bit confused on how to manipulate definitions and theorems in finding counterexamples. Take the following lemma

Lemma 13.1 Let $X$ be a set; let $\beta$ be a basis for a topology $\tau$ on $X$. Then $\tau $ equals the collection of all unions of elements of $\beta$.

So In this case how would I properly manipulate/remove the hypothesis. For some reason I just end up getting Let X be a set; Then $\tau$ equals the collection of all unions of elements of $\beta$.

Sorry if this sounds stupid and ridiculous but I just need clarification.

How would I in general manipulate definitions/theorems?

Answer 1

You could rephrase the particular statement you wrote as follows.

Let $X$ be a set and $\tau$ a topology on $X$. Assume also that $\beta\subseteq\tau$ is a basis for $\tau$. Then given $U\in\tau$ there is a subset $\mathcal{V}\subseteq\tau$ such that $\bigcup_{V\in\mathcal{V}}=U$.

Depending on what definition of basis you are using this statement may be trivial. For example if your definition of a basis is:

For a topology $\tau$ on a set $X$ a subset $\beta\subseteq\tau$ is a basis for $\tau$ provided that every element of $\tau$ is a union of elements in $\beta$.

Then of course the statement is trivial. However if your definition of basis reads something like:

For a topology $\tau$ on a set $X$ a subset $\beta\subseteq\tau$ is a basis for $\tau$ provided provided that for every $U\in\tau$ and $x\in U$ there is a $V\in\beta$ for which $x\in V\subseteq U$.

Then there is something (minor) to be said about why such a definition would imply that $\tau$ is made up of unions of elements of $\beta$.

In general, when dissecting mathematical statements it is important to recall (or look up) what the definitions of the key terms are. Then what a statement is "saying" usually becomes more clear.