@MISC{TOPS_5.MIZ,
  AUTHOR = {Koch, Sebastian},
  NOTE1 = {The author is enrolled in the Johannes Gutenberg University in Mayence, Germany},
  TITLE = {Some Remarks about Product Spaces},
  SECTION1 = {Preliminaries},
  SECTION2 = {Remarks about Product Spaces},
  DAY = {29},
  MONTH = {September},
  YEAR = {2018},
  ADDRESS1 = {Germany\\\href{mailto:skoch02@students.uni-mainz.de}{skoch02@students.uni-mainz.de}},
  SUMMARY = {This article covers some technical aspects about the product topology 
   which are usually not given much of a thought in mathematics and standard literature 
  like \cite{MUNKRES} and \cite{KELLEY}, not even by Bourbaki in \cite{BOUR}. 
  Let $\{\mathcal T_i\}_{i\in I}$ be a family of topological spaces. The   
  prebasis of the product space $\mathcal T=\prod_{i\in I} \mathcal T_i$ is 
  defined in \cite{WAYBEL18.ABS} as the set of all $\pi^{-1}_i(V)$ with $i\in I$ 
  and $V$ open in $\mathcal T_i$. Here it is shown that the   
  basis generated by this prebasis consists exactly of the sets $\prod_{i\in I} V_i$ 
  with $V_i$ open in $ \mathcal T_i$ and for all but finitely many 
  $i\in I$ holds $V_i=\mathcal T_i$. Given $I=\{a\}$ we have $  \mathcal T\cong \mathcal T_a$, 
  given $I=\{a,b\}$ with $a\neq b$ we have $\mathcal T \cong \mathcal T_a   
  \times\mathcal T_b$. Given another family of topological spaces 
  $\{\mathcal S_i\}_{i\in I}$ such that $\mathcal S_i\cong\mathcal T_i$ 
  for all $i\in I$, we have $\mathcal S=\prod_{i\in I} \mathcal S_i\cong   
  \mathcal T$. If instead $S_i$ is a subspace of $T_i$ for each $i\in I$, 
  then $\mathcal S$ is a subspace of $\mathcal T$. These results are 
  obvious for mathematicians, but formally proven here.}}

@BOOK{BOUR,
  TITLE = {Elements of mathematics},
  VOLUME = {Pt. 1: 3, General topology},
  AUTHOR = {Bourbaki, Nicolas},
  ADDRESS = {Paris},
  PUBLISHER = {Hermann [u.a.]},
  YEAR = {(1966)},
  PAGES = {VIII, 436 pages}
}

@BOOK{MUNKRES,
  TITLE = {Topology},
  AUTHOR = {Munkres, James Raymond},
  ADDRESS = {Upper Saddle River, NJ},
  PUBLISHER = {Prentice-Hall},
  YEAR = {2000},
  EDITION = {2. ed.},
  PAGES = {XVI, 537 pages}
}

@BOOK{KELLEY,
  TITLE = {General topology},
  SERIES = {University series in higher mathematics},
  AUTHOR = {Kelley, John L.},
  ADDRESS = {Princeton, NJ [u.a.]},
  PUBLISHER = {Van Nostrand},
  YEAR = {1963},
  EDITION = {Reprint},
  PAGES = {XIV, 298 pages},
}