let X be non empty TopSpace; ( X is locally-relatively-compact implies X is locally-closed/compact )
assume A1:
X is locally-relatively-compact
; X is locally-closed/compact
let x be Point of X; COMPACT1:def 5 ex U being a_neighborhood of x st
( U is closed & U is compact )
consider U being a_neighborhood of x such that
A2:
U is relatively-compact
by A1;
A3:
x in Int U
by CONNSP_2:def 1;
Int U c= Int (Cl U)
by PRE_TOPC:18, TOPS_1:19;
then
Cl U is a_neighborhood of x
by A3, CONNSP_2:def 1;
hence
ex U being a_neighborhood of x st
( U is closed & U is compact )
by A2; verum