#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 \[ \overline{S}:=\text{cl}(S):=\text{ad}(S):=\bigcap\{C\in{\cal C}_{{\cal T}}\mid S\subseteq C\} \] \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 \[ \mathring{S}:=\text{int}S:=\bigcup\{A\in{\cal T}\mid A\subseteq S\} \] \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