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+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Section
+Clausura
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $(X,{\cal T})$
+\end_inset
+
+ un espacio topológico y 
+\begin_inset Formula $S\subseteq X$
+\end_inset
+
+, la 
+\series bold
+clausura
+\series default
+ o 
+\series bold
+adherencia
+\series default
+ de 
+\begin_inset Formula $S$
+\end_inset
+
+ es el menor cerrado que contiene a 
+\begin_inset Formula $S$
+\end_inset
+
+, es decir, la intersección de todos los cerrados que lo contienen, y se
+ denota 
+\begin_inset Formula 
+\[
+\overline{S}:=\text{cl}(S):=\text{ad}(S):=\bigcap\{C\in{\cal C}_{{\cal T}}:S\subseteq C\}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dado 
+\begin_inset Formula $p\in X$
+\end_inset
+
+, 
+\begin_inset Formula $p\in\overline{S}\iff\forall V\in{\cal E}(p),V\cap S\neq\emptyset$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $p\in\overline{S}$
+\end_inset
+
+ y supongamos que existe 
+\begin_inset Formula $V\in{\cal E}(p)$
+\end_inset
+
+ con 
+\begin_inset Formula $V\cap S=\emptyset$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $S\subseteq X\backslash V\in{\cal C_{T}}$
+\end_inset
+
+, luego 
+\begin_inset Formula $p\in\overline{S}\subseteq X\backslash V$
+\end_inset
+
+.
+\begin_inset Formula $\#$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $p\in X$
+\end_inset
+
+ tal que 
+\begin_inset Formula $V\cap S\neq\emptyset\forall V\in{\cal E}(x)$
+\end_inset
+
+ y supongamos 
+\begin_inset Formula $p\notin\overline{S}$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $p\in X\backslash\overline{S}\in{\cal E}(p)$
+\end_inset
+
+, pero 
+\begin_inset Formula $(X\backslash\overline{S})\cap S=\emptyset$
+\end_inset
+
+.
+\begin_inset Formula $\#$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $(X,d)$
+\end_inset
+
+ es un espacio métrico y 
+\begin_inset Formula $S\subseteq X$
+\end_inset
+
+, dado 
+\begin_inset Formula $p\in X$
+\end_inset
+
+, 
+\begin_inset Formula $p\in\overline{S}\iff d(p,S)=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $p\in\overline{S}$
+\end_inset
+
+, si suponemos 
+\begin_inset Formula $d(p,S)=r>0$
+\end_inset
+
+, entonces 
+\begin_inset Formula $B(p;r)\cap S=\emptyset$
+\end_inset
+
+, lo que contradice 
+\begin_inset Formula $p\in\overline{S}$
+\end_inset
+
+.
+\begin_inset Formula $\#$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $d(p,S)=0$
+\end_inset
+
+, 
+\begin_inset Formula $\forall n\in\mathbb{N},\exists q\in S:d(p,q)<\frac{1}{n}$
+\end_inset
+
+, luego 
+\begin_inset Formula $\forall n\in\mathbb{N},B(p;\frac{1}{n})\cap S\neq\emptyset$
+\end_inset
+
+ y 
+\begin_inset Formula $p\in\overline{S}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $S\subseteq T\implies\overline{S}\subseteq\overline{T}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $S\subseteq T\subseteq\overline{T}\in{\cal C_{T}}$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $\overline{T}$
+\end_inset
+
+ es un cerrado que contiene a 
+\begin_inset Formula $S$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $\overline{S}\subseteq\overline{T}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\bigcup_{i\in I}\overline{S_{i}}\subseteq\overline{\bigcup_{i\in I}S_{i}}$
+\end_inset
+
+; 
+\begin_inset Formula $\bigcup_{i=1}^{n}\overline{S_{i}}=\overline{\bigcup_{i=1}^{n}S_{i}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\subseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $\forall j\in I,S_{j}\subseteq\bigcup_{i\in I}S_{i}\implies\overline{S_{j}}\subseteq\overline{\bigcup_{i\in I}S_{i}}\implies\bigcup_{i\in I}\overline{S_{i}}\subseteq\overline{\bigcup_{i\in I}S_{i}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\supseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $\overline{\bigcup_{i\in I}S_{i}}\subseteq\overline{\bigcup_{i\in I}\overline{S_{i}}}\overset{\text{\textbf{SI \ensuremath{I} es finito}}}{=}\bigcup_{i\in I}\overline{S_{i}}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $\overline{\bigcap_{i\in I}S_{i}}\subseteq\bigcap_{i\in I}\overline{S_{i}}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\forall i\in I,S_{i}\subseteq\overline{S_{i}}\implies\bigcap_{i\in I}S_{i}\subseteq\bigcap_{i\in I}\overline{S_{i}}\implies\overline{\bigcap_{i\in I}S_{i}}\subseteq\overline{\bigcap_{i\in I}\overline{S_{i}}}=\bigcap_{i\in I}\overline{S_{i}}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $S\in{\cal C_{T}}\iff\overline{S}=S$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $S\in{\cal C_{T}}\implies\overline{S}\subseteq S\overset{S\subseteq\overline{S}}{\implies}S=\overline{S}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $S=\overline{S}\in{\cal C_{T}}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $\overline{\overline{S}}=\overline{S}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $D\subseteq X$
+\end_inset
+
+ es 
+\series bold
+denso
+\series default
+ en 
+\begin_inset Formula $(X,{\cal T})$
+\end_inset
+
+ si 
+\begin_inset Formula $\overline{D}=X$
+\end_inset
+
+, si y sólo si cualquier abierto no vacío corta a 
+\begin_inset Formula $D$
+\end_inset
+
+.
+ 
+\begin_inset Formula $(X,{\cal T})$
+\end_inset
+
+ es 
+\series bold
+separable
+\series default
+ si admite un subconjunto denso y numerable.
+\end_layout
+
+\begin_layout Standard
+Todo espacio numerable es separable pero el recíproco no se cumple, pues
+ por ejemplo, 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ es denso en 
+\begin_inset Formula $(\mathbb{R},{\cal T}_{u})$
+\end_inset
+
+ y numerable y por tanto 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ es separable, pero no es numerable.
+ Igualmente 
+\begin_inset Formula $(X,{\cal T}_{D})$
+\end_inset
+
+ es separable si y sólo si es numerable, mientras que 
+\begin_inset Formula $(X,{\cal T}_{CF})$
+\end_inset
+
+ es siempre separable (basta tomar un subconjunto numerable no finito).
+\end_layout
+
+\begin_layout Section
+Puntos de acumulación y aislados
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $S\subseteq X$
+\end_inset
+
+, 
+\begin_inset Formula $p\in X$
+\end_inset
+
+ es un 
+\series bold
+punto de acumulación
+\series default
+ de 
+\begin_inset Formula $S$
+\end_inset
+
+ si 
+\begin_inset Formula $\forall U\in{\cal E}(p),(U\backslash\{p\})\cap S\neq\emptyset$
+\end_inset
+
+.
+ Llamamos 
+\series bold
+acumulación
+\series default
+ o 
+\series bold
+conjunto derivado
+\series default
+ de 
+\begin_inset Formula $S$
+\end_inset
+
+ (
+\begin_inset Formula $\text{ac}(S)$
+\end_inset
+
+ o 
+\begin_inset Formula $S'$
+\end_inset
+
+) al conjunto de todos los puntos de acumulación de 
+\begin_inset Formula $S$
+\end_inset
+
+.
+ Por otro lado, 
+\begin_inset Formula $p\in S$
+\end_inset
+
+ es un 
+\series bold
+punto aislado
+\series default
+ de 
+\begin_inset Formula $S$
+\end_inset
+
+ si 
+\begin_inset Formula $\exists U\in{\cal E}(p):U\cap S=\{p\}$
+\end_inset
+
+, y el conjunto de todos los puntos aislados de 
+\begin_inset Formula $S$
+\end_inset
+
+ es 
+\begin_inset Formula $\text{ais}(S)=S\backslash S'$
+\end_inset
+
+, y se tiene que 
+\begin_inset Formula $\overline{S}=S\cup S'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Frontera
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $S\subseteq X$
+\end_inset
+
+, 
+\begin_inset Formula $p\in X$
+\end_inset
+
+ es un 
+\series bold
+punto frontera
+\series default
+ de 
+\begin_inset Formula $S$
+\end_inset
+
+ si 
+\begin_inset Formula $\forall U\in{\cal E}(p),(U\cap S\neq\emptyset\land U\cap(X\backslash S)\neq\emptyset)$
+\end_inset
+
+.
+ Llamamos 
+\series bold
+frontera
+\series default
+ de 
+\begin_inset Formula $S$
+\end_inset
+
+ (
+\begin_inset Formula $\partial S$
+\end_inset
+
+ o 
+\begin_inset Formula $\text{fr}(S)$
+\end_inset
+
+) al conjunto de todos los puntos frontera de 
+\begin_inset Formula $S$
+\end_inset
+
+.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\partial S=\overline{S}\cap\overline{X\backslash S}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\partial S\in{\cal C_{T}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Además, en un espacio métrico,
+\begin_inset Formula 
+\begin{eqnarray*}
+p\in\partial S & \iff & \forall r>0,(B(p;r)\cap S\neq\emptyset\land B(p;r)\cap(X\backslash S)\neq\emptyset)\\
+ & \iff & \forall n\in\mathbb{N},(B(p;\frac{1}{n})\cap S\neq\emptyset\land B(p;\frac{1}{n})\cap(X\backslash S)\neq\emptyset)\\
+ & \iff & d(p,S)=d(p,X\backslash S)=0
+\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Interior
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $(X,{\cal T})$
+\end_inset
+
+ un espacio topológico y 
+\begin_inset Formula $S\subseteq X$
+\end_inset
+
+, el 
+\series bold
+interior
+\series default
+ de 
+\begin_inset Formula $S$
+\end_inset
+
+ es el mayor abierto contenido en 
+\begin_inset Formula $S$
+\end_inset
+
+, es decir, la unión de todos los abiertos contenidos en 
+\begin_inset Formula $S$
+\end_inset
+
+, y se denota
+\begin_inset Formula 
+\[
+\mathring{S}:=\text{int}S:=\bigcup\{A\in{\cal T}:A\subseteq S\}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Newpage newpage
+\end_inset
+
+Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\mathring{S}=X\backslash\overline{X\backslash S}$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\subseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $p\in\mathring{S}\implies\exists A\in{\cal T}:p\in A\subseteq\mathring{S}\subseteq S\implies A\cap(X\backslash S)=\emptyset\implies p\notin\overline{X\backslash S}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\supseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $\begin{array}{c}
+X\backslash S\subseteq\overline{X\backslash S}\implies X\backslash\overline{X\backslash S}\subseteq S\\
+X\backslash\overline{X\backslash S}\in{\cal T}
+\end{array}\implies X\backslash\overline{X\backslash S}\subseteq\mathring{S}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $S\in{\cal T}\iff S=\mathring{S}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\partial S=\overline{S}\backslash\mathring{S}$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+\partial S=\overline{S}\cap\overline{X\backslash S}=\overline{S}\cap(X\backslash\mathring{S})=\overline{S}\backslash\mathring{S}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $S\in{\cal T}\iff S\cap\partial S=\emptyset$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $S\in{\cal T}\implies\partial S=\overline{S}\backslash\mathring{S}=\overline{S}\backslash S\implies\partial S\cap S=\emptyset$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $\emptyset=\partial S\cap S=(\overline{S}\backslash\mathring{S})\cap S=S\backslash\mathring{S}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $p\in\mathring{S}\iff\exists U\in{\cal E}(p):U\subseteq S$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $S\subseteq T\implies\mathring{S}\subseteq\mathring{T}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\bigcap_{i=1}^{n}\mathring{S_{i}}=\mathring{\overbrace{\bigcap_{i=1}^{n}S_{i}}}$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+\begin{array}{c}
+\mathring{S}\cap\mathring{T}=(X\backslash\overline{X\backslash S})\cap(X\backslash\overline{X\backslash T})=X\backslash(\overline{X\backslash S}\cup\overline{X\backslash T})=\\
+=X\backslash\overline{(X\backslash S)\cup(X\backslash T)}=X\backslash\overline{X\backslash(S\cap T)}=\mathring{\overbrace{S\cap T}}
+\end{array}
+\]
+
+\end_inset
+
+Esto NO se cumple para la unión.
+\end_layout
+
+\begin_layout Standard
+Además, en un espacio métrico,
+\begin_inset Formula 
+\begin{eqnarray*}
+p\in\mathring{S} & \iff & \exists r>0:B(p;r)\subseteq S\\
+ & \iff & \exists n\in\mathbb{N}:B(p;\frac{1}{n})\subseteq S\\
+ & \iff & d(p,X\backslash S)>0
+\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Clausura, frontera e interior relativos
+\end_layout
+
+\begin_layout Standard
+Escribimos 
+\begin_inset Formula $\text{cl}_{X}(S)$
+\end_inset
+
+, 
+\begin_inset Formula $\text{int}_{X}(S)$
+\end_inset
+
+ y 
+\begin_inset Formula $\partial_{X}(S)$
+\end_inset
+
+ en 
+\begin_inset Formula $(X,{\cal T})$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{cl}_{H}(S)$
+\end_inset
+
+, 
+\begin_inset Formula $\text{int}_{H}(S)$
+\end_inset
+
+ y 
+\begin_inset Formula $\partial_{H}(S)$
+\end_inset
+
+ en 
+\begin_inset Formula $(H,{\cal T}|_{H})$
+\end_inset
+
+.
+ Así, sea 
+\begin_inset Formula $(X,{\cal T})$
+\end_inset
+
+ un espacio topológico y 
+\begin_inset Formula $S\subseteq H\subseteq X$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{cl}_{H}(S)=\text{cl}_{X}(S)\cap H$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\subseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sabemos que 
+\begin_inset Formula $S\subseteq\text{cl}_{X}(S)\cap H\in{\cal C}_{H}$
+\end_inset
+
+, y como 
+\begin_inset Formula $\text{cl}_{H}(S)$
+\end_inset
+
+ es el menor cerrado en 
+\begin_inset Formula $H$
+\end_inset
+
+ que contiene a 
+\begin_inset Formula $S$
+\end_inset
+
+, 
+\begin_inset Formula $\text{cl}_{H}(S)\subseteq\text{cl}_{X}(S)\cap H$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\supseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $p\in\text{cl}_{X}(S)\cap H$
+\end_inset
+
+ y 
+\begin_inset Formula $U'\in{\cal E}_{H}(p)$
+\end_inset
+
+, entonces existe 
+\begin_inset Formula $U\in{\cal E}_{X}(p)$
+\end_inset
+
+ tal que 
+\begin_inset Formula $U'=U\cap H$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $p\in\text{cl}_{X}(S)$
+\end_inset
+
+, 
+\begin_inset Formula $U\cap S\neq\emptyset$
+\end_inset
+
+, ahora bien, 
+\begin_inset Formula $U'\cap S=U\cap H\cap S=U\cap S\neq\emptyset$
+\end_inset
+
+, luego 
+\begin_inset Formula $p\in\text{cl}_{H}(S)$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $\text{int}_{X}(S)\cap H\subseteq\text{int}_{H}(S)$
+\end_inset
+
+, y esta inclusión suele ser estricta.
+\series bold
+
+\begin_inset Newline newline
+\end_inset
+
+
+\series default
+
+\begin_inset Formula $\text{int}_{X}(S)\cap H$
+\end_inset
+
+ es un abierto de 
+\begin_inset Formula $H$
+\end_inset
+
+ contenido en 
+\begin_inset Formula $S$
+\end_inset
+
+, y por tanto 
+\begin_inset Formula $\text{int}_{X}(S)\cap H\subseteq\text{int}_{H}(S)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\partial_{H}(S)\subseteq\partial_{X}(S)\cap H$
+\end_inset
+
+.
+\begin_inset Formula 
+\begin{multline*}
+\begin{array}{c}
+\partial_{H}(S)=\text{cl}_{H}(S)\backslash\text{int}_{H}(S)\subseteq(\text{cl}_{X}(S)\cap H)\backslash(\text{int}_{X}(S)\cap H)=\\
+=(\text{cl}_{X}(S)\backslash\text{int}_{X}(S))\cap H=\partial_{X}(S)\cap H
+\end{array}
+\end{multline*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Convergencia
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $\{x_{n}\}_{n=1}^{\infty}$
+\end_inset
+
+ una sucesión de puntos de 
+\begin_inset Formula $X$
+\end_inset
+
+, 
+\begin_inset Formula $\{x_{n}\}_{n=1}^{\infty}$
+\end_inset
+
+ 
+\series bold
+converge
+\series default
+ o 
+\series bold
+tiende
+\series default
+ a 
+\begin_inset Formula $x$
+\end_inset
+
+ (
+\begin_inset Formula $x_{n}\rightarrow x$
+\end_inset
+
+ o 
+\begin_inset Formula $\lim x_{n}=x$
+\end_inset
+
+) si 
+\begin_inset Formula $\forall U\in{\cal E}(x),\exists n_{U}\in\mathbb{N}:\forall n\geq n_{U},x_{n}\in U$
+\end_inset
+
+.
+ En particular, en un espacio métrico 
+\begin_inset Formula $(X,d)$
+\end_inset
+
+, 
+\begin_inset Formula $x_{n}\rightarrow x\iff\forall\varepsilon>0,\exists n_{\varepsilon}\in\mathbb{N}:\forall n\geq n_{\varepsilon},x_{n}\in B(x;r)$
+\end_inset
+
+, o lo que es lo mismo, si la sucesión 
+\begin_inset Formula $\{d(x_{n},x)\}_{n=1}^{\infty}$
+\end_inset
+
+ converge a 0 en 
+\begin_inset Formula $(\mathbb{R},d_{u})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $(X,d)$
+\end_inset
+
+ un espacio métrico, 
+\begin_inset Formula $S\subseteq X$
+\end_inset
+
+ y 
+\begin_inset Formula $x\in X$
+\end_inset
+
+, entonces 
+\begin_inset Formula $x\in\overline{S}\iff\exists\{x_{n}\}_{n=1}^{\infty}\subseteq S:x_{n}\rightarrow x$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $x\in\overline{S}$
+\end_inset
+
+, para cada 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+, 
+\begin_inset Formula $B(x;\frac{1}{n})\cap S\neq\emptyset$
+\end_inset
+
+, luego podemos tomar 
+\begin_inset Formula $x_{n}\in B(x;\frac{1}{n})\cap S$
+\end_inset
+
+ y construir así la sucesión.
+ Entonces 
+\begin_inset Formula $d(x_{n},x)<\frac{1}{n}$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $x_{n}\rightarrow x$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Cualquier 
+\begin_inset Formula $U\in{\cal E}(x)$
+\end_inset
+
+ contiene puntos de la sucesión, de forma que 
+\begin_inset Formula $U\cap S\neq\emptyset$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $x\in\overline{S}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Así pues, en un espacio métrico 
+\begin_inset Formula $(X,d)$
+\end_inset
+
+, 
+\begin_inset Formula $S$
+\end_inset
+
+ es denso en 
+\begin_inset Formula $X$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $\forall x\in X,\exists\{x_{n}\}_{n=1}^{\infty}\subseteq S:x_{n}\rightarrow x$
+\end_inset
+
+, y 
+\begin_inset Formula $x\in\partial S$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $\exists\{x_{n}\}_{n=1}^{\infty}\subseteq S,\{y_{n}\}_{n=1}^{\infty}\subseteq X\backslash S:x_{n},y_{n}\rightarrow x$
+\end_inset
+
+.
+ Estas caracterizaciones sólo son ciertas en espacios métricos, pero no
+ es espacios topológicos arbitrarios.
+\end_layout
+
+\end_body
+\end_document