I already searched and saw some questions about definitions of locally compact spaces (like Two definitions of locally compact space), but I could NOT find the exact comparison of the two definitions I want to discuss below:
Definition 1: Let T=(S,τ) be a topological space. Then T is locally compact iff every point of S has a local basis $\mathscr B$ such that all elements of $\mathscr B$ are compact. (from https://proofwiki.org/wiki/Definition:Locally_Compact)
Definition 2: Let T=(S,τ) be a topological space. Then T is locally compact iff - a) T is Hausdorff; and b) Every point has a compact neighbourhood (from https://www.math.ksu.edu/~nagy/real-an/1-05-top-loc-comp.pdf)
I could see that from Definition 1, it is easy to get property b of Definition 2
From https://proofwiki.org/wiki/Definition:Locally_Compact, it comments that "if T is a Hausdorff space, then Definition 1 and Definition 2 are equivalent" - could anyone help to prove the direction from Definition 2 to Definition 1 when T is a Hausdorff space?