Answer by Anton Fetisov for Defining topological spaces with the notion of...
No such definition can be given for a general topological space, because a general topological space can have very few morphisms from or into $[0;1]$. E.g. consider $\Bbb Q$: there are no non-trivial...
View ArticleAnswer by Ronnie Brown for Defining topological spaces with the notion of...
Try the wikipedia entry on sequential spaces. I quote:"Many conditions have been shown to be equivalent to $X$ being sequential. Here are a few:$X$ is the quotient of a first countable space.$X$ is the...
View ArticleAnswer by Mike Shulman for Defining topological spaces with the notion of...
I don't think I've seen a definition of a space-like notion phrased only in terms of paths, but you could certainly write one down, perhaps as an example of concrete sheaves. It seems related to the...
View ArticleDefining topological spaces with the notion of continuous path
Let’s consider the following pseudo-definition of (nice) topological spaces : a space is a set $X$ together with distinguished paths $[0,1]\to{}X$ called continuous paths, distinguished maps...
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