Sir, around 39:00 you say that in topological spaces, everything is defined in terms of open sets, while mentioning earlier that all topological spaces are not induced by a metric. So how does one define an open set without using a metric? Is there an even more general definition of open sets?
Yes, there is. In a more general setting, one can define a subset to be "open" if it satisfies three properties. A family of open subsets of the base set is called a "topology", and the base set together with a topology is called a "topological spaces". Then it can be shown that the so-called "open sets" defined by means of a metric also satisfy these properties, meaning that metric spaces are a subclass of topological spaces, i.e. all metrics "induce" a topology on the base set. It is non-trivial, however, to show that there are topologies that *cannot* be induced by a metric PS beware that it is false that "all top. spaces are not induced by a metric". The correct statement would be "not all top. spaces are induced by a metric".
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Thank you sir
Sir, around 39:00 you say that in topological spaces, everything is defined in terms of open sets, while mentioning earlier that all topological spaces are not induced by a metric. So how does one define an open set without using a metric? Is there an even more general definition of open sets?
Yes, there is. In a more general setting, one can define a subset to be "open" if it satisfies three properties. A family of open subsets of the base set is called a "topology", and the base set together with a topology is called a "topological spaces". Then it can be shown that the so-called "open sets" defined by means of a metric also satisfy these properties, meaning that metric spaces are a subclass of topological spaces, i.e. all metrics "induce" a topology on the base set. It is non-trivial, however, to show that there are topologies that *cannot* be induced by a metric
PS beware that it is false that "all top. spaces are not induced by a metric". The correct statement would be "not all top. spaces are induced by a metric".
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