Understanding Point Types in Real Analysis

[STEMucator]

Just a few definitions I would like to verify so I'm not studying the wrong stuff.

Interior Point : A point $Q \in S \subseteq ℝ^n$ is an interior point of S if $\forall \delta > 0, \exists N_{\delta}(Q) \subseteq S$. The interior of S consists of all interior points and is denoted $S˚$

Boundary Point : A point $Q \in S \subseteq ℝ^n$ is a boundary point of S if $\forall \delta > 0, \exists P_1 \in S \wedge P_2 \in (ℝ^n - S) \space| \space P_1, P_2 \in N_{\delta}(Q)$

Limit Point : A point $Q \in S \subseteq ℝ^n$ is a limit point of S if $\forall \delta > 0, \exists P \in S \space | \space P \in N_{\delta}(Q), \space P≠Q$

Trying to condense my stuff, hopefully I'm doing this correctly.

 

[micromass]

Just a few definitions I would like to verify so I'm not studying the wrong stuff.

Interior Point : A point $Q \in S \subseteq ℝ^n$ is an interior point of S if $\forall \delta > 0, \exists N_{\delta}(Q) \subseteq S$. The interior of S consists of all interior points and is denoted $S˚$

It should be $\exists \delta >0$.

Boundary Point : A point $Q \in S \subseteq ℝ^n$ is a boundary point of S if $\forall \delta > 0, \exists P_1 \in S \wedge P_2 \in (ℝ^n - S) \space| \space P_1, P_2 \in N_{\delta}(Q)$

Limit Point : A point $Q \in S \subseteq ℝ^n$ is a limit point of S if $\forall \delta > 0, \exists P \in S \space | \space P \in N_{\delta}(Q), \space P≠Q$

OK. But these definitions also hold if you don't demand $Q\in S$.

 

[STEMucator]

It should be $\exists \delta >0$.



OK. But these definitions also hold if you don't demand $Q\in S$.

I thought that it was for all deltas I choose? What if I chose delta so large that my neighborhood was contained in the compliment?

Also, for the purposes of a calc II course, we're assuming Q is inside of the set, so I should be good there for now.