branch of mathematics, a subset E {\displaystyle E} of a topological space X {\displaystyle X} is said to be locally closed if any of the following equivalent...

taken for locally compact, countably compact, metrizable, separable, countable; the second for locally connected. Locally closed subset A subset of a topological...

not a complete subset). This shows that in a Hausdorff locally convex space that is not complete, the closed convex hull of compact subset might fail to...

a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A...

examples show that a subset of a locally compact space need not be locally compact, which contrasts with the open and closed subsets in the previous section...

compact Hausdorff space is locally compact, a connected space—and even a connected subset of the Euclidean plane—need not be locally connected (see below)...

{\displaystyle A} is a clopen subset of Q . {\displaystyle \mathbb {Q} .} ( A {\displaystyle A} is not a clopen subset of the real line R {\displaystyle...

{\displaystyle f:X\to Y} maps closed subsets of its domain to closed subsets of its codomain; that is, for any closed subset C {\displaystyle C} of X , {\displaystyle...

Cartesian closed, then it is called locally cartesian closed. Note that if C is locally Cartesian closed, it need not actually be Cartesian closed; that happens...

{\displaystyle \Gamma _{x}\subset \Gamma '_{x}} where the equality holds if X {\displaystyle X} is compact Hausdorff or locally connected. A space in which...