@BalarkaSen okay good point so for vector spaces it's about obstructions to isomorphisms of linear maps, yet for homological algebra you have to generalize this and say it's an obstruction to something:
"we measure the obstruction to some construction. This is the basis of homological algebra: (co)homology groups, toda classes, chern classes, thom classes and so on."
https://www.physicsforums.com/threads/vector-field-uniquely-determined-by-rot-div.174451/#post-1363006
(post 6)
which explains why concepts such as $\mathrm{Ext}$ and $\mathrm{Tor}$ have an obstruction interpretation

Hello @TedShifrin !!

I am reading a proof and there is the following part:

"Since $\gamma$ is regular, at least one of $\dot u(t_0)$, $\dot v(t_0)$ and $\dot w(t_0)$ is non-zero, say $\dot u(t_0)$. This means that the graph of $u$ as a function of $t$ is not parallel to the $t$-axis at $t_0$.
This implies that any line parallel to the $t$-axis close to $u=x_0$ intersects the graph of $u$ at a unique point $u(t)$ with $t$ close to $t_0$."

Could you explain to me the part "This implies that any line parallel to the $t$-axis close to $u=x_0$ intersects the graph of $u$ at a unique point $u(…

@TedShifrin So suppose that $x,y \in R^{\omega}$ such that x - y is eventually zero, note x and y can be writtten as $x = (x_1,....,x_i,...)$, $y = (y_1,....,y_i,....)$. By hypothesis there exists $N \in \mathbb{N}$ such that $x_i - y_i = 0 \forall i > N$, so that means that $x_i = y_i \forall i > N$
$f : R^n \rightarrow R^{\omega}$ where $(q_1,q_2,...,q_n) \mapsto (q_1,q_2,...,q_n,x_{N + 1},x_{N + 2},....)$
such map is continous
and $x,y$ is in $f(R^n)$
since R^n is connected so is the image, but the image is then a component of x,y.