# Proofs of Theorems in Topology

We prove some theorems in topology using the fewest number of symbols at each step. Our purpose is pedagogical.

https://doi.org/10.31219/osf.io/wn24y

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 4, Article 267, 2022 [KM]

# Theorems in Topology

This is an introductory collection of theorems in topology.

https://doi.org/10.31219/osf.io/zm56w

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 4, Article 260, 2022 [KE]

# Metrizable Topological Space

[knowledge base]

METRIZABLE TOPOLOGICAL SPACE and its underlying definitions are presented in this white paper (knowledge base).

https://doi.org/10.31219/osf.io/f8vez

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 205, 2021 [IB]

# Presheaf (of abelian groups) on a topological space

[knowledge base]

PRESHEAF and its underlying definitions are presented in this white paper (knowledge base).

https://doi.org/10.31219/osf.io/2y5s4

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 201, 2021 [HY]

# Generating finitely many circles from one circle using Banach-Tarski decomposition paradox

[microreview]

We prove the Banach-Tarski decomposition paradox applied to a circle.

https://doi.org/10.31219/osf.io/mh5gx

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 2, Article 142, 2020 [FN]

# A hole in the black hole

[microresearch]

Supposedly, matter falls inside the black hole whenever it reaches its event horizon. The Planck scale, however, imposes a limit on how much matter can occupy the center of a black hole. It is shown here that the density of matter exceeds Planck density in the singularity, and as a result, spacetime tears apart. After the black hole is formed, matter flows from its center to its border due to a topological force; namely, the increase on the tear of spacetime due to its limit, until it reaches back to the event horizon, generating the firewall phenomenon. We conclude that there is no spacetime inside black holes. We propose a solution to the black hole information paradox.

https://doi.org/10.31219/osf.io/js7rf

Open Physics Collaboration

Op. J. Math. Phys.
Volume 2, Article 78, 2020 [DA]

# Dark matter and bubbles of void

[original insight]

This is a thought experiment with the following conjecture: bubbles of spacetime void alter the gravitational field of the galaxy and might explain the dark matter phenomenon.

https://doi.org/10.31219/osf.io/w7m3q

Open Physics Collaboration

Op. J. Math. Phys.
Volume 1, Article 33, 2019 [BG]

# Route one to quantum gravity

[original idea]

These are the first steps, I believe, that can contribute to the understanding of the most fundamental nature of spacetime.

https://doi.org/10.31219/osf.io/4mbxq

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 1, Article 16, 2019 [AP]

[original idea]