Tim Maudlin, a philosopher of science at NYU, has a book out called: New Foundations for Physical Geometry: The Theory of Linear Structures.
The section on about the book says the following:
Topology is the mathematical study of the most basic geometrical structure of a space. Mathematical physics uses topological spaces as the formal means for describing physical space and time. This book proposes a completely new mathematical structure for describing geometrical notions such as continuity, connectedness, boundaries of sets, and so on, in order to provide a better mathematical tool for understanding space-time... The Theory of Linear Structures replaces the foundational notion of standard topology, the open set, with the notion of a continuous line.
The last line in the quote above caught my eye and a cursory reading of one of the chapters (7: Metrical Structures) on Google Books set off a few alarms. But, I am far from a mathematical physicist and my searches of reviews were fruitless, so my question is:
Is there a link to a review of this book or what is the considered opinion about it among mathematical physicist?
edit 1 I'll understand if this is closed. I suddenly realised that I am effectively indulging in what our friends in the sociology department love to call 'policing the boundaries' of our science.
edit 2 Looking at how things are, I'd vote to close this question too if I could (without intending any offence to those who have participated in the discussion). However, the discussion below made me peek superficially into the history of things. Evidently, the foundations of pont-set topology as we understand it now was established by the 1920s, born from considerations in analysis, it began with Frechet's 1904 thesis, where he based an abstraction of the euclidean space on the concept of limits. It is interesting to note that Ricci and Levi-Civita's Methods de calcul differential absolu et leurs applications was published in 1901 for work done in the previous decade.