A locally closed subset of $\mathbb R^2$?

Question

Is this set a locally closed one in $\mathbb R^2$ with standard topology?

$$\{ (x,y)| y>0\text{ or }x=0 \}$$

thnks.

Answer 1

You can use the characterisation given in your previous question:

A subset $A$ of a topological space $X$ is locally closed iff $A$ is an open subset of $\overline{A}$ (in the usual subspace topology).

Letting $$A = \{ \langle x , y \rangle \in \mathbb{R} \times \mathbb{R} : y > 0 \text{ or } x = 0 \},$$ it is not difficult to show that $\overline{A} = \{ \langle x,y \rangle \in \mathbb{R} \times \mathbb{R} : y \geq 0\text{ or }x = 0 \}$.

Note that any open neighbourhood of $\langle 0,0 \rangle$ (in $\mathbb{R} \times \mathbb{R}$) must contain a point of the form $\langle x , 0 \rangle$ where $x \neq 0$.