Düzgün Denk Metrikler-I

Question

$(X,d_1),(X,d_2)$ metrik uzaylar olmak üzere

$$d_1\overset{D}{\sim}d_2$$

$$\Leftrightarrow$$

$$(\forall\epsilon>0)(\exists\delta_1,\delta_2>0)(\forall x\in X)[B^{d_1}(x,\delta_1)\subseteq B^{d_2}(x,\epsilon)\wedge B^{d_2}(x,\delta_2)\subseteq B^{d_1}(x,\epsilon)]$$

olduğunu gösteriniz.

Tanım: $(X,d_1),(X,d_2)$ metrik uzaylar olmak üzere

$$d_1\overset{D}{\sim}d_2$$

$$:\Leftrightarrow$$

$$(\forall\epsilon>0)(\exists\delta_1,\delta_2>0)(\forall x,y\in X)$$ $$[(d_1(x,y)<\delta_1\Rightarrow d_2(x,y)<\epsilon)\wedge (d_2(x,y)<\delta_2\Rightarrow d_1(x,y)<\epsilon)]$$

Answer 1

İspat:

$$d_1\overset{D}{\sim}d_2$$

$$\Leftrightarrow$$

$$(\forall\epsilon>0)(\exists\delta_1,\delta_2>0)(\forall x,y\in X)[(d_1(x,y)<\delta_1\Rightarrow d_2(x,y)<\epsilon)\wedge (d_2(x,y)<\delta_2\Rightarrow d_1(x,y)<\epsilon)]$$

$$\Leftrightarrow$$

$$(\forall\epsilon>0)(\exists\delta_1,\delta_2>0)(\forall x,y\in X)[(y\in B^{d_1}(x,\delta_1)\Rightarrow y\in B^{d_2}(x,\epsilon))\wedge (y\in B^{d_2}(x,\delta_2)\Rightarrow y\in B^{d_1}(x,\epsilon))]$$

$$\Leftrightarrow$$

$$(\forall\epsilon>0)(\exists\delta_1,\delta_2>0)(\forall x\in X)[B^{d_1}(x,\delta_1)\subseteq B^{d_2}(x,\epsilon)\wedge B^{d_2}(x,\delta_2)\subseteq B^{d_1}(x,\epsilon)].$$