Styjun

Give an example of a space which is locally compact at all but one point.

My professor told me that

Wedge product of infinitely many circles

Will do the job. But I'm not getting his answer. Other examples will be appreciated too.

A variation of the wedge space that your professor suggested: the hedgehog of infinite "spininess":

Let $X = [0,1] times mathbbR$, where we identify all points $(0,t)$ to a single point (equivalence class, really, we coudl call it plain $0$, the origin) with the metric:

$$dleft((x,t), (x',t')right) =
begincases
|x-x'| & text if t=t' \
|x| + |x'| & text if t neq t' \
endcases$$

so the space looks like $mathbbR$ many copies of $[0,1]$ glued together at their $0$, metrically. If does not matter what representation $(0,t)$ we choose for $0$ in this computation (so the metric is well-defined).

At all points $(x,t)$ different from $0$ the space behave locally just like $(0,1]$ so is locally compact, but no neighbourhood of $0$ is compact: for some ball neighbourhood $B(0,r)$ for some $r>0$ pick $0 < s < r$ and note that $(s,t): t in mathbbR$ is an infinite closed and discrete subset of $B(0,r)$ showing its non-compactness. So I could have sufficed with index set $mathbbN$ in the second coordinate for this...)

For a nice overview at an elementary level of such "hedgehog spaces", see this paper, e.g. They can be quite useful.

HINT: In slightly more elementary terms what your professor is suggesting is the following.

Consider infinitely many (a numerable infinity will suffice) copies $C_1$, $C_2$, $C_3$, ... of the circle $S^1$ with its natural topology and on each copy fix a point
$$
P_iin C_i.
$$
Now consider the quotient
$$
X=left.bigcup_iC_iright/sim
$$
where the only non trivial equivalence is $P_1sim P_2sim P_3simcdots$. Let $Pin X$ be the class of the points $P_i$. What is a neighborhood of $P$?

(Added later)

More generally you can consider a numerable family of pointed locally compact spaces $T_i, P_iin T_i_iinBbb N$ and form the quotient space
$$
X=left.bigcup_iinBbb NT_iright/sim
$$
where the $P_i$ are declared equivalent to each other, and $Pin X$ their common class. Then $X$ is locally compact at each of its points $Qneq P$ but $P$ doesn't have a compact neighborhood since every open set $Pin Usubset X$ is of the form
$$
U=left.bigcup_iinBbb NA_iright/sim
$$
with $A_i$ open in $T_i$ and $A_icap A_j=P$ for every $ineq j$.

Here is a different example. Take the universal cover of a disk in $mathbbR^2$: $widetildeB(0,1)−0$. Endow it with the Euclidean metric it inherits from the disk, and then take its completion with respect to that metric.