1

Prove the set $S=\left\{ \left(x,y\right);ax+by<c\right\} $ is open

My Approach I know there are many methods of proving this.But i find the method of proving every point to be interior point very fundamental.Please help me with this method only.

Let $A=\left(x_{o},y_{o}\right)$$\in S$ Now i need to prove that a Ball $B\left(A,r\right)$$\subset$$S$. I am unable to find any Ball.

Then i Saw Book's Approach It says

$\left(x_{o},y_{o}\right)$$\in S$ $\Longrightarrow$$ax_{o}+by_{o}<c$$\Longrightarrow$I Don't Understand how $ax_{o}+by_{o}<c$$\Longrightarrow$$\delta<\frac{|ax_{o}+by_{o}-c|}{\sqrt{a^{2}+b^{2}}}$

I know it is delta neighbourhood around $\left(x_{o},y_{o}\right)$.But how did they get this expression

and how can we say N$_{\delta}$$\left(A\right)$$\subset$S

Mohan Sharma
  • 1,009

1 Answers1

1

Let $\varphi : \Bbb R^2 \to \Bbb R$ be given by $(x,y) \mapsto ax+by$. It is a linear combination of the two standard projection maps, so it is continuous, so the pre-image of any open set is an open set, and in particular $S = \varphi^{-1}[(-\infty,c)]$ is the preimage of the open interval $(-\infty,c)$, so it is open.

Kenny Lau
  • 25,049
  • is this theorem work proving for closed sets ? – Mohan Sharma Jan 20 '18 at 09:33
  • The preimage of a closed set under a continuous function is a closed set. See here for a topological proof. – Kenny Lau Jan 20 '18 at 09:35
  • Sir please explain how $ax_{o}+by_{o}<c$$\Longrightarrow$$\delta<\frac{|ax_{o}+by_{o}-c|}{\sqrt{a^{2}+b^{2}}}$ – Mohan Sharma Jan 20 '18 at 09:37
  • See this. – Kenny Lau Jan 20 '18 at 09:38
  • Sharks.Kenny.Would like to have your take.Assume a,b,c positive. First quadrant the line ax+by -c=0 cuts a triangle, x intercept c/a, y intercept b/a. Choose any point (x_0,y_0) within triangle, i.e. below the line , I.e. ax0 +by_0 < c. Distance from this point to line is given by the quoted formula, call it d. Consider the circle around (x_0,y_0)with radius r=d/2 (for example). All points within the open "ball" are within triangle, I.e. ax_0+by_0 <c. Visually clear. Do not manage to prove this without some coordinate geometry .Any ideas? – Peter Szilas Jan 20 '18 at 18:27