Locally compact space versus locally connected space

Question

I remember when I was studying the notion of local connectedness, it meant that each point has "arbitrarily small" neighborhoods that are connected.

More precisely, one has the following definition:

Definition: A space $X$ is said to be locally connected at $x$ if for every neighborhood $U$ of $x$, there is a connected neighborhood $V$ of $x$ contained in $U$.

But right now when I am studying the definition of locally compact space, I thought that the definition should be the following: a space $X$ is said to be locally compact at $x$ if for every neighborhood $U$ of $x$, there is a compact neighborhood $K$ of $x$ contained in $U$.

However the definition in the book is a bit different: A space $X$ is said to be locally compact at $x$ if there is some compact subspace $C$ of $X$ that contains a neighborhood of $x$.

1. Can anyone explain to me please why there is such a distinction between definition of local connected space and local compact space?

2. And the definition in the book says "..contains a neighborhood of $x$". Does it mean that for any neighborhood $U$ of $x$, there is a compact subspace $C$ of $x$ such that $x\in U\subset C$. Am I right?

Answer 1

As to 2: $C$ contains a neighbourhood of $x$ means that there is an open subset $U$ containing $x$ (this is what neighbourhood of $x$ means in your text, I gather) such that $U \subseteq C$ ($C$ contains $U$).

If you allow for neighbourhoods to be non-open you could more succinctly say that "$C$ is a compact neighbourhood of $x$".

And yes there is a subtle distinction in the "local" versions of connectedness and compactness. In practice, spaces are often Hausdorff ($T_2$) and then a locally compact (in your definition) Hausdorff space indeed obeys the stronger

for every open neighbourhood $U$ of $x$ there is a compact neighbourhood $C$ of $x$ inside it, or equivalently some open $V$ with $x \in V \subseteq \overline{V} \subseteq U$ with $\overline{V}$ compact.

And the latter property is what's most often used in proofs.

The simple definition of having a compact neighbourhood (or compact set containing an open neighbourhood) has the advantage that it's immediately clear that compact spaces are also locally compact (just take $C=X$ everywhere), and locally compact is meant as generalisation/loosening of compactness, whereas for connectedness it's meant as an extra refinement (a space can be connected and not locally connected, e.g.), so the requirements are stricter. The local compactness in your variant is easier to check and Hausdorffness will make it strong and more useful (in analysis e..g.) as in the highlighted version.