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So after comparing the definitions, I have no idea why these seem to be defined so differently.

A space $X$ at $x$ is locally connected if there is an open set $U$ containing $x$ such that there we can find a open connected set $x \in V \subset U$. I guess in other words, the open connected sets $V_x$ form a local base.

A space $X$ at $x$ is locally compact if there is a compact neighbourhood $C$ at $x$.

It looks like to me that local compactness is roughly saying a space $X$ is locally compact on some point $x$, if on some neighbourhood $U(x)$, $X$ is compact on that neighbourhood. In other words, $U(x) \cap X = U(x)$ is compact, which seems to agree with what one might normally come up with for the definition.

On the other hand, local connectedness says we can find a connected neighbourhood inside an open set...

I've read some answers from MO, but none of these feels satisfying to me (a lot of red-herring answers I feel...). I just don't understand what's wrong with defining local connectedness as having "a connected neighbourhood" or local compactness as "a compact set inside an open set". Maybe there are spaces that simply do not have connected (or compact) subsets?

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Normally you would define local compactness like this:

For every point $x$ and a neighbourhood $x\in U$ there is a subneighbourhood $x\in V\subset U$ such that $\overline{V}$ is compact.

The thing is that these two definitions of local compactness are equivalent. Indeed, if $U$ is relatively compact (i.e. $\overline{U}$ is compact) and $V$ is any other subset then $U\cap V$ is again relatively compact.

The analogy does not work for connectedness. The intersection of a connected open subset with any other open subset does not have to be connected (even when both are connected).

Let's define:

  • C: connected
  • LC: locally connected (with the usual definition)
  • LC2: every point has a connected neighbourhood

Then C implies LC2. Indeed, if $X$ is connected then every point has a connected neighbourhood, namely $X$. Also LC implies LC2.

On the other hand neither LC implies C (discrete space with at least two points) nor C implies LC (the topologist's sine curve).

As for why LC is used instead of LC2, well it's because "necessity is the mother of invention".

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More precisely, locally compact means has a base of open sets with compact closures. A weaker definition is: has a base of open sets each of which is contained in a compact set. For Hausdorff spaces, the definitions are equivalent. Thus is a definition in the form of has a base of open sets that are whatever; like locally connected - has a base of open connected sets.

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