In the definition of a locally connected space we demand every neighbourhood of a point to satisfy certain condition whereas for a locally compact space we demand that one neighbourhood be there with the required property. 


Is there some reason for this difference? Is it so that a compact space needs(?) to be locally compact?


Connectedness is a "geometric" property whereas compactness is an "analytic" property. Is that a reason behind such different definitions?



locally connected versus locally compact
1. One could phrase local compactness using neighbourhood bases as well (in the Hausdorff case, at least) if desired: once one has one precompact open neighbourhood, one automatically has a whole neighbourhood base of precompact sets, since any subset of a precompact set is still precompact. (And in practice, this is often how local compactness is actually *used*.)
2. The most important thing that a property named "locally P" should obey is that it be *local* rather than global. In the case of topological spaces, this means that if a topological space X is locally P at some point $x\_0$, and we have another topological space Y which agrees with X at a neighbourhood of $x\_0$ (e.g. Y could be the restriction of X to a neighbourhood of $x\_0$, with the relative topology), then Y should be locally P at $x\_0$ as well. Both local compactness and local connectedness, as defined traditionally, have this locality property. On the other hand, the property "$x\_0$ has at least one connected neighbourhood" is not local (consider for instance $({\bf Q} \times {\bf R}) \cup \{\infty\}$ in the Riemann sphere ${\bf R}^2 \cup \{\infty\}$, which is a globally connected space which is locally identical near the origin to ${\bf Q} \times {\bf R}$ in ${\bf R}^2$, which has no connected open sets), and thus this property does not deserve the name of "local connectedness at $x\_0$".


It may help to think of the modifier "locally" not as a rigid recipe for converting global properties to local ones, but rather as an indicator that the property being modified is a local analogue of the global, unmodified, property. In most cases there is only one obvious such analogue to select, although in some cases (such as the notion of "locally compact" in the non-Hausdorff setting) there is some freedom of choice, which ultimately means that one has make a somewhat arbitrary convention regarding terminology at some point.


