Confusion about definition of locally compact topological space

Question

I was reading van Dantzig's theorem from Terry Tao's notes, where I encountered that he's writing locally compact topological space to mean every point has a compact neighborhood.

What does locally compact mean? Every point having compact neighborhood or every point having neighborhood whose closure is compact?

If anyone explains in detail it would be great.

Answer 1

$X$ is a topological space throughout.

There are two standard definitions of a neighborhood. Both are used widely. The difference you are highlighting comes down to whether neighborhoods are defined to contain an open subset, or defined to be an open subset.

• For $x \in X$, $V \subset X$ is a neighborhood(1) of $x$ if $x \in U \subset V$ for some open set $U \subset X$. (note 1)
• For $x \in X$, $V \subset X$ is a neighborhood(2) of $x$ if $x \in V$ and $V$ is open. (note 2)

In the familiar setting $X = \mathbb{R}$, there are no neighborhood(2)s that are compact. However, for each $x \in X$, there are neighborhood(2)s, $V_x = (x-1/2, x+1/2)$, for instance, whose closure is closed and bounded, hence compact (by the Heine-Borel theorem). In this setting, the closures of the neighborhood(2)s are neighborhood(1)s, so we get these two characterizations:

• $\mathbb{R}$ is locally compact because every $x \in \mathbb{R}$ has a compact neighborhood(1).
• $\mathbb{R}$ is locally compact because every $x \in \mathbb{R}$ has a neighborhood(2) whose closure is compact.

(note 1): See, for instance, Bredon, Glen E., Topology and Geometry, ISBN ISBN 0-387-97926-3, p. 4: "If $X$ is a topological space and $x \in X$ then a set $N$ is called a neighborhood of $x$ in $X$ if there is an open set $U \subset N$ with $x \in U$."

(note 2): See, for instance, Munkres, James R., Topology: A First Course, ISBN 0-13-925495-1, p. 96: "Mathematicians often use some special terminology here. They shorten the statement '$U$ is an open set containing $x$' to the phrase '$U$ is a neighborhood of $x$.'" This language also appears in the second edition of this work, ISBN 978-0131816299, still p. 96. Alternatively, Kelley, John L., General Topology, ISBN 0-387-90125-6, p. 14: "A neighborhood of a point is any open set containing this point." Kelley then asserts that analysts and Bourbaki use "neighborhood" in the neighborhood(1) sense.