What does the closure of a set mean?

Question

In my book for projective geometry, this symbol: < x > means a subspace containing points x. But my teacher calls it "the closure of x". Does this mean the same thing. He also described "closure operations". Is the closure just talking about the subspace formed by the set of points still just like my book?

Answer 1

In general, given any operatino $*$, a set $S$ has a closure if $*$ when operated on elements of $S$ (the # of elements of $S$ on which $*$ is operated depends on $*$ itself) gives elements which belong to $S$ itself. I think your teacher called $\langle x\rangle$ the $\bf{closure}$ of $x$ because here we have a Vector space and thus operations $+$ and $.$ on it, so wrt to these operation on $\langle x\rangle$ has closure property.