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| -rw-r--r-- | library/everything.tex | 1 | ||||
| -rw-r--r-- | library/topology/urysohn2.tex | 19 |
2 files changed, 20 insertions, 0 deletions
diff --git a/library/everything.tex b/library/everything.tex index b966197..94dd6d8 100644 --- a/library/everything.tex +++ b/library/everything.tex @@ -30,6 +30,7 @@ \import{topology/disconnection.tex} \import{topology/separation.tex} \import{numbers.tex} +\import{topology/urysohn2.tex} \begin{proposition}\label{trivial} $x = x$. diff --git a/library/topology/urysohn2.tex b/library/topology/urysohn2.tex index ea49a6c..a64fa7e 100644 --- a/library/topology/urysohn2.tex +++ b/library/topology/urysohn2.tex @@ -16,6 +16,22 @@ \section{Urysohns Lemma} +\begin{definition}\label{one_to_n_set} + $\seq{m}{n} = \{x \in \naturals \mid m \leq x \leq n\}$. +\end{definition} + +\begin{struct}\label{sequence} + A sequence $X$ is a onesorted structure equipped with + \begin{enumerate} + \item $\index$ + \item $\indexset$ + \end{enumerate} + such that + \begin{enumerate} + \item\label{indexset_is_subset_naturals} $\indexset[X] \subseteq \naturals$. + \item\label{index_is_bijection} $\index[X]$ is a bijection from $\indexset[X]$ to $\carrier[X]$. + \end{enumerate} +\end{struct} \begin{abbreviation}\label{urysohnspace} $X$ is a urysohn space iff @@ -33,6 +49,7 @@ + \begin{theorem}\label{urysohn} Let $X$ be a urysohn space. Suppose $A,B \in \closeds{X}$. @@ -52,6 +69,8 @@ % & x ,x \in X <- will result in technicly ambigus parse \end{cases} + + $U_1$ Trivial. |