In my Algebraic Topology course we are studying CW complexes and at some point my teacher gave us a non-example I do not totally understand, let me explain.
Consider the space $X = \bigcup_n X_n$ where $X_n$ is the circle in $\mathbb R^2$ of radius $1/n$ centered at $(1/n, 0)$, which should look like something like this:
He told us this was not a CW complex but did not tell us why. Clearly it may be constructed by attaching a countable number of circles to a point. The only axiom in the definition of CW complex that is likely to be problematic in this case is (in my opinion)
A subset $A \subset X$ is closed in $X$ if and only if $A \cap X^n$ is closed in $X^n$ for all $n$.
But still I am unable to construct an closed set in $X$ that is not closed in $X^n$, I am not totally confortable with the topology on CW complexes. Could one of you help me with this ?
It's the right-to-left direction that causes the problem (a closed subset of $X$ is automatically a closed subset in any subspace of $X$ that happens to contain it). To see what goes wrong in the left-to-right direction, look at the subset $A = \{ (2/n, 0) : n = 1, 2, \ldots\}$. It isn't closed in $X$, because its limit point $(0, 0)$ is not in $A$, but its intersection with any $X^n$ is finite and hence closed.