I have studied that topologically equivalent metrics produce the same open and closed sets. They also produce same compact and connected subsets. Does it mean that topologically equivalent metrics have same open, closed or compact subsets? In what context topologically equivalent metrics differ from each other?
Intutively, how can we explain topologically equivalent metrics? Can we say that two topologically equivalent metrics have same topological properties?
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On
The difference isn't topological because the two metric space are equal up homeomorfism. But it can exist many difference of orther type. For example the euclidean plane and the hyperbolic plane are equivalents topologically, as metric spaces, but geometrically they are not essentially equivalent.
On
In its simplest form, the fact that two metrics $d_1$ and $d_2$ are topologically equivalent means that given any point $x$ and a ball around $x$ in one of the metrics, it contains a ball around $x$ in the other.
That means that topological equivalence of metrics is a local condition; it only cares about small distances. Topological equivalence also means that any property that can be defined with open and / or closed sets is invariand under change of metric. This includes compactness, clopenness, convergence of sequenses, and continuity of functions in and out of the space.
Properties more closely related to the global properties of the metric, like uniform continuity are not , in general, kept intact.
Suppose that $d_0$ and $d_1$ are metrics on a set $X$. Each generates a topology: $d_0$ generates $$\tau_0=\left\{V\subseteq X:\forall x\in V\exists \epsilon_x>0\big(B_{d_0}(x,\epsilon_x)\subseteq V\big)\right\}\;,$$ and $d_1$ generates $$\tau_1=\left\{V\subseteq X:\forall x\in V\exists \epsilon_x>0\big(B_{d_1}(x,\epsilon_x)\subseteq V\big)\right\}\;,$$ where $B_{d_0}(x,\epsilon)=\{y\in X:d_0(x,y)<\epsilon\}$ and $B_{d_1}(x,\epsilon)=\{y\in X:d_1(x,y)<\epsilon\}$. We say that $d_0$ and $d_1$ are topologically equivalent iff $\tau_0=\tau_1$: the two metrics generate the same topology. Thus, the metric spaces $\langle X,d_0\rangle$ and $\langle X,d_1\rangle$ share all purely topological properties, because they are identical as topological spaces: they are both $\langle X,\tau\rangle$, where $\tau=\tau_0=\tau_1$. Thus, they have the same open sets, closed sets, compact sets, connected sets, discrete sets, etc.
They do not necessarily share specifically metric properties. For example, one of $\langle X,d_0\rangle$ and $\langle X,d_1\rangle$ may be complete and the other not. One may be bounded and the other not. One may be totally bounded and the other not. They may differ in anything that depends specifically on the metric, rather than on the topology.