We raise here the following question: “can a vector space \(E\) be written as a finite union of proper subspaces”?

Let’s consider the simplest case, i.e. writing \(E= V_1 \cup V_2\) as a union of two proper subspaces. By hypothesis, one can find two non-zero vectors \(v_1,v_2\) belonging respectively to \(V_1 \setminus V_2\) and \(V_2 \setminus V_1\). The relation \(v_1+v_2 \in V_1\) leads to the contradiction \(v_2 = (v_1+v_2)-v_1 \in V_1\) while supposing \(v_1+v_2 \in V_2\) leads to the contradiction \(v_1 = (v_1+v_2)-v_2 \in V_2\). Therefore, a vector space can never be written as a union of two proper subspaces.

We now analyze if a vector space can be written as a union of \(n \ge 3\) proper subspaces. We’ll see that it is impossible when \(E\) is a vector space over an infinite field. But we’ll describe a counterexample of a vector space over the finite field \(\mathbb{Z}_2\) written as a union of three proper subspaces. Continue reading A vector space written as a finite union of proper subspaces→

I’m used to publish mathematical counterexamples. But following January 7th attack in Paris against “Charlie Hebdo” journalists, my mind is busy with other topics.

Despite the willingness of some zealots, freedom of speech is and will remain alive.

We recall some definitions on open and closed maps. In topology an open map is a function between two topological spaces which maps open sets to open sets. Likewise, a closed map is a function which maps closed sets to closed sets.

For a continuous function \(f: X \mapsto Y\), the preimage \(f^{-1}(V)\) of every open set \(V \subseteq Y\) is an open set which is equivalent to the condition that the preimages of the closed sets (which are the complements of the open subsets) in \(Y\) are closed in \(X\). However, a continuous function might not be an open map or a closed map as we prove in following counterexamples. Continue reading Continuous maps that are not closed or not open→