Topological and Metric Properties

[Bachelier]

Can someone explain the difference between the two?

2 topo spaces are isometric if they have the same metric properties and homeometric if they have the same topological properties.

If a space is homeo it is iso, but not vice verse. Which begs the conclusion that every topological property is a metric one, but not every metric is topological.
Is it that a property is metric if it is related to the metric used on the space. That's how in the same space like R, we can prove that cauchiness is not topological by changing the metric. So is Cauchiness a metric property? What about Boundedness?
However continuity of a function is topological. Meaning it is mainly linked to the space we're working in?

I'd appreciate some input and especially examples of different metric and topological properties.

 

[HallsofIvy]

"metric" spaces form a subset of all topological spaces- that is a metric space is a topological space in which the topology is defined by a specfic metric. But not all topological spaces are metric spaces. It is NOT true that "if a space is homeo then it is iso". In fact, it doesn't even make sense to talk about a space being "homeo" or "iso". Two spaces may be homeomorphic or isomorphic to one another.

As for "every topological property is a metric one but not every metric (property) is topological". Yes, a metric space is a topological space so every property defined for all topological spaces applies to metric spaces also. No, not every topological space admits a metric so some properties of metric spaces (such as "bounded") applies to general topological spaces. If, by "Cauchiness" (a word I hope I never use again!) you mean "the Cauchy Criterion", that is, necessarily a metric property since you require that $|a_n- a_m|$ go to 0 which requires a metric.

Note that "compactness" is a topological property but "boundedness" is not so it only makes sense to ask if all compact sets are bounded in metric spaces.

 

[lavinia]

Can someone explain the difference between the two?

2 topo spaces are isometric if they have the same metric properties and homeometric if they have the same topological properties.

If a space is homeo it is iso, but not vice verse. Which begs the conclusion that every topological property is a metric one, but not every metric is topological.
Is it that a property is metric if it is related to the metric used on the space. That's how in the same space like R, we can prove that cauchiness is not topological by changing the metric. So is Cauchiness a metric property? What about Boundedness?
However continuity of a function is topological. Meaning it is mainly linked to the space we're working in?

I'd appreciate some input and especially examples of different metric and topological properties.

Two spaces can be homeo without being iso.

 

[Bachelier]

I get it. Because if 2 homeo TPs have a different metric, then they won't be iso.

Thanks