No topologies characterize differentiability as continuity

Abstract

Do there exist topologies U \mathcal {U} and V \mathcal {V} for the set R R of real numbers such that a function f f from R R to R R is smooth in some specified sense (e.g., differentiable, C n {C^n} , or C ∞ {C^\infty } ) with respect to the usual structure of the real line if and only if f f is continuous from U \mathcal {U} to V \mathcal {V} ? We show that the answer is no.

Testing modified gravity and no-hair relations for the Kerr-Newman metric through quasiperiodic oscillations of galactic microquasars

“…To ensure that the spacetime M is asymptotically flat, one requires the existence of a manifold S, consisting of S plus one additional point Λ, subject to certain conditions [8,20]. This point Λ is to be thought of as the 'asymptotic' point, in the sense that we can extend the metric h to the boundary of S. In particular, we require the existence of a scalar field, Ω such that hab = Ω 2 h ab is a metric on S, and at Λ one has Ω = 0, Da Ω = 0, and Da Db Ω = n hab for some n. Euclidean 3-space is asymptotically flat in the above sense with conformal factor Ω = r −2 , where r is the distance from some origin [8,9,20].…”

“…In this paper, we explore the properties of this metric as a solution to a vacuum f (R) theory. We introduce multipole moments for higher-order curvature theories through the Geroch-Hansen procedure, [3,8,9,20,21] compute the Kerr-Newman moments, and find that they reduce to their Kerr counterparts found by Hansen, [9] when the appropriate limit is taken.…”

The causal boundary of product spacetimes

“…The first attempt to establish identifications in ∂(V ) ∪ ∂(V ) was proposed in [4]. The authors introduced a generalized Alexandrov topology on V ♮ : the topology generated by the sub-basis…”

“…However, this method is neither systematic nor intrinsic, and sometimes it results very restrictive. In order to overcome these handicaps, Geroch, Kronheimer and Penrose introduced a new construction called causal boundary [4]. In this new approach they attach a future (past) ideal point for every inextensible, physically admissible future (past) trajectory, in such a way that the ideal point only depends on the past (future) of the trajectory.…”

“…Many authors have tried different methods to establish these identifications [4,5,6,7]; indeed, this question is related to the introduction of a satisfactory topology for the completion. However, they have not obtained totally satisfactory results up to date (see [8,9,10,1]).…”