TOPOLOGY
$Let\; X\; be\; non \;empty\; set\; and\; \tau\; be\; the\; family\; of\; subsets\; of\; X.\;$
$ We\; say\; \tau\; is\; topology\; on\; X,\; if\; and\; only\; if$
$Let\; X\; be\; non \;empty\; set\; and\; \tau\; be\; the\; family\; of\; subsets\; of\; X.\;$
$ We\; say\; \tau\; is\; topology\; on\; X,\; if\; and\; only\; if$
$1. \;\phi\;, X\; \in \;\tau$
$2. \;Arbitrary\; union\; of\; members\; of\; \tau\; belongs\; to\; \tau$
$3. \;Finite\; intersection\; of\; members\; of\; \tau\; belongs\; to\; \tau$
NOTE:
NOTE:
$\star\; X\; is\; called\; ground\; set\; and\; its\; elements\; are\; called\; points.\;$
$\star\; The\; sets\; in\; \tau\; are\; open\; sets.\;$
$\star \;Union\; of\; two\; toplogical\; space\; may\; or\; may\; form\; toplogical\; space$
$\star \;Intersection\; of\; two\; toplogical\; space\; always\;form\; toplogical\; space$
Examples:
$\star \;Union\; of\; two\; toplogical\; space\; may\; or\; may\; form\; toplogical\; space$
$\star \;Intersection\; of\; two\; toplogical\; space\; always\;form\; toplogical\; space$
Examples:
$1.\;P(X)=\tau\; [Discrete\; Topology/Largest\; Topology ]$
$2.\;\tau=\{\phi,X\}\; [Indiscrete\; Topology/Smallest \;Topology ]$
$3.\;[Cofinite\; Topological\; space]$
$\bullet\tau=\{\phi\; and\; subsets\; of\; X\; whose\; Complements\; are\; finite\} $
$\bullet\tau=\{\phi\; and\; subsets\; of\; X\; whose\; Complements\; are\; finite\} $
$4.\;[Cocountable\; Topological\; space]$
$\bullet\tau=\{\phi\; and\; subsets\; of\; X\; whose\; Complements\; are\; countable\}\;$
$\bullet\tau=\{\phi\; and\; subsets\; of\; X\; whose\; Complements\; are\; countable\}\;$
$5.\;Usual\; Topological\; Space$
$\bullet \tau=\{Union\; of\; open\; intervals\; in\; \mathbb{R}\}$
$\bullet\tau=\{Union\; of\; open\; Discs\; in\; \mathbb{R^2}\}$
$\bullet\tau=\{Union\; of\; open\; spheres\; in\; \mathbb{R^3}\}$
$6.\; Upper\; Limit\; Topology$
$\bullet\tau=\{Unions\; of\; open-closed\; intervals\; (a,b]\}$
$7.\; Lower\; Limit\; Topology$
$\bullet\tau=\{Unions\; of\; closed-open\; intervals\; [a,b)\}$
Coarser/Weaker Topological Space:
$Let\;\tau_{1},\;\tau_{2}\;be\;two\;topological\;spaces\;and\;if\;\tau_{1}\subseteq \tau_{2}\;$
$then\;\tau_{1}\;is\;said\;to\;be\;Coarser/Weaker\;topological space\;$
Finer/Stronger Topological Space:
$Let\;\tau_{1},\;\tau_{2}\;be\;two\;topological\;spaces\;and\;if\;\tau_{1}\subseteq \tau_{2}\;$
$ then\;\tau_{2}\;is\;said\;to\;be\;Finer/Stronger\;topological space.$
Incomparable Topological Spaces
$Let\;\tau_{1},\;\tau_{2}\;be\;two\;topological\;spaces\;and\;if\;neither\;\tau_{1}\nsubseteq \tau_{2}\;nor$
$\;\tau_{2}\nsubseteq \tau_{1}\; then\;\tau_{1},\;\tau_{2}\;are\;Incomparable\;topological\; spaces.\;$
Relative Topology
$If\; (X,\tau)\; is\; topological\; space\; and\; A\subseteq X\; $
$Let\; \tau_{A}=\{A\cup U:U\in \tau\}\; is\;relative\;topology\;and\;(A,\tau_{A})\;is\;topological\;subspace.$
![TOPOLOGY $Let\; X\; be\; non \;empty\; set\; and\; \tau\; be\; the\; family\; of\; subsets\; of\; X.\;$ $ We\; say\; \tau\; is\; topology\; on\; X,\; if\; and\; only\; if$ $1. \;\phi\;, X\; \in \;\tau$ $2. \;Arbitrary\; union\; of\; members\; of\; \tau\; belongs\; to\; \tau$ $3. \;Finite\; intersection\; of\; members\; of\; \tau\; belongs\; to\; \tau$ NOTE: $\star\; X\; is\; called\; ground\; set\; and\; its\; elements\; are\; called\; points.\;$ $\star\; The\; sets\; in\; \tau\; are\; open\; sets.\;$ $\star \;Union\; of\; two\; toplogical\; space\; may\; or\; may\; form\; toplogical\; space$ $\star \;Intersection\; of\; two\; toplogical\; space\; always\;form\; toplogical\; space$ Examples: $1.\;P(X)=\tau\; [Discrete\; Topology/Largest\; Topology ]$ $2.\;\tau=\{\phi,X\}\; [Indiscrete\; Topology/Smallest \;Topology ]$ $3.\;[Cofinite\; Topological\; space]$ $\bullet\tau=\{\phi\; and\; subsets\; of\; X\; whose\; Complements\; are\; finite\} $ $4.\;[Cocountable\; Topological\; space]$ $\bullet\tau=\{\phi\; and\; subsets\; of\; X\; whose\; Complements\; are\; countable\}\;$ $5.\;Usual\; Topological\; Space$ $\bullet \tau=\{Union\; of\; open\; intervals\; in\; \mathbb{R}\}$ $\bullet\tau=\{Union\; of\; open\; Discs\; in\; \mathbb{R^2}\}$ $\bullet\tau=\{Union\; of\; open\; spheres\; in\; \mathbb{R^3}\}$ $6.\; Upper\; Limit\; Topology$ $\bullet\tau=\{Unions\; of\; open-closed\; intervals\; (a,b]\}$ $7.\; Lower\; Limit\; Topology$ $\bullet\tau=\{Unions\; of\; closed-open\; intervals\; [a,b)\}$ Coarser/Weaker Topological Space: $Let\;\tau_{1},\;\tau_{2}\;be\;two\;topological\;spaces\;and\;if\;\tau_{1}\subseteq \tau_{2}\;$ $then\;\tau_{1}\;is\;said\;to\;be\;Coarser/Weaker\;topological space\;$ Finer/Stronger Topological Space: $Let\;\tau_{1},\;\tau_{2}\;be\;two\;topological\;spaces\;and\;if\;\tau_{1}\subseteq \tau_{2}\;$ $ then\;\tau_{2}\;is\;said\;to\;be\;Finer/Stronger\;topological space.$ Incomparable Topological Spaces $Let\;\tau_{1},\;\tau_{2}\;be\;two\;topological\;spaces\;and\;if\;neither\;\tau_{1}\nsubseteq \tau_{2}\;nor$ $\;\tau_{2}\nsubseteq \tau_{1}\; then\;\tau_{1},\;\tau_{2}\;are\;Incomparable\;topological\; spaces.\;$ Relative Topology $If\; (X,\tau)\; is\; topological\; space\; and\; A\subseteq X\; $ $Let\; \tau_{A}=\{A\cup U:U\in \tau\}\; is\;relative\;topology\;and\;(A,\tau_{A})\;is\;topological\;subspace.$](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjpGe9DnwI5WcbhBYHbolqcVSlQfDnhctLSDiMdBNdHjmurnXpcE3SLsVCJOT8kKn22l2PVN2X1UbujSGjcnCpyLfT1a1GyjpQ4QGWPtVi-HAkPGqRneL22179Fav8Jp8Belwnv2-1H5dx/s640/topology-1.png "Topology Short Notes 1")
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Coarser/Weaker Topological Space:
$Let\;\tau_{1},\;\tau_{2}\;be\;two\;topological\;spaces\;and\;if\;\tau_{1}\subseteq \tau_{2}\;$
$then\;\tau_{1}\;is\;said\;to\;be\;Coarser/Weaker\;topological space\;$
Finer/Stronger Topological Space:
$Let\;\tau_{1},\;\tau_{2}\;be\;two\;topological\;spaces\;and\;if\;\tau_{1}\subseteq \tau_{2}\;$
$ then\;\tau_{2}\;is\;said\;to\;be\;Finer/Stronger\;topological space.$
Incomparable Topological Spaces
$Let\;\tau_{1},\;\tau_{2}\;be\;two\;topological\;spaces\;and\;if\;neither\;\tau_{1}\nsubseteq \tau_{2}\;nor$
$\;\tau_{2}\nsubseteq \tau_{1}\; then\;\tau_{1},\;\tau_{2}\;are\;Incomparable\;topological\; spaces.\;$
Relative Topology
$If\; (X,\tau)\; is\; topological\; space\; and\; A\subseteq X\; $
$Let\; \tau_{A}=\{A\cup U:U\in \tau\}\; is\;relative\;topology\;and\;(A,\tau_{A})\;is\;topological\;subspace.$
![TOPOLOGY $Let\; X\; be\; non \;empty\; set\; and\; \tau\; be\; the\; family\; of\; subsets\; of\; X.\;$ $ We\; say\; \tau\; is\; topology\; on\; X,\; if\; and\; only\; if$ $1. \;\phi\;, X\; \in \;\tau$ $2. \;Arbitrary\; union\; of\; members\; of\; \tau\; belongs\; to\; \tau$ $3. \;Finite\; intersection\; of\; members\; of\; \tau\; belongs\; to\; \tau$ NOTE: $\star\; X\; is\; called\; ground\; set\; and\; its\; elements\; are\; called\; points.\;$ $\star\; The\; sets\; in\; \tau\; are\; open\; sets.\;$ $\star \;Union\; of\; two\; toplogical\; space\; may\; or\; may\; form\; toplogical\; space$ $\star \;Intersection\; of\; two\; toplogical\; space\; always\;form\; toplogical\; space$ Examples: $1.\;P(X)=\tau\; [Discrete\; Topology/Largest\; Topology ]$ $2.\;\tau=\{\phi,X\}\; [Indiscrete\; Topology/Smallest \;Topology ]$ $3.\;[Cofinite\; Topological\; space]$ $\bullet\tau=\{\phi\; and\; subsets\; of\; X\; whose\; Complements\; are\; finite\} $ $4.\;[Cocountable\; Topological\; space]$ $\bullet\tau=\{\phi\; and\; subsets\; of\; X\; whose\; Complements\; are\; countable\}\;$ $5.\;Usual\; Topological\; Space$ $\bullet \tau=\{Union\; of\; open\; intervals\; in\; \mathbb{R}\}$ $\bullet\tau=\{Union\; of\; open\; Discs\; in\; \mathbb{R^2}\}$ $\bullet\tau=\{Union\; of\; open\; spheres\; in\; \mathbb{R^3}\}$ $6.\; Upper\; Limit\; Topology$ $\bullet\tau=\{Unions\; of\; open-closed\; intervals\; (a,b]\}$ $7.\; Lower\; Limit\; Topology$ $\bullet\tau=\{Unions\; of\; closed-open\; intervals\; [a,b)\}$ Coarser/Weaker Topological Space: $Let\;\tau_{1},\;\tau_{2}\;be\;two\;topological\;spaces\;and\;if\;\tau_{1}\subseteq \tau_{2}\;$ $then\;\tau_{1}\;is\;said\;to\;be\;Coarser/Weaker\;topological space\;$ Finer/Stronger Topological Space: $Let\;\tau_{1},\;\tau_{2}\;be\;two\;topological\;spaces\;and\;if\;\tau_{1}\subseteq \tau_{2}\;$ $ then\;\tau_{2}\;is\;said\;to\;be\;Finer/Stronger\;topological space.$ Incomparable Topological Spaces $Let\;\tau_{1},\;\tau_{2}\;be\;two\;topological\;spaces\;and\;if\;neither\;\tau_{1}\nsubseteq \tau_{2}\;nor$ $\;\tau_{2}\nsubseteq \tau_{1}\; then\;\tau_{1},\;\tau_{2}\;are\;Incomparable\;topological\; spaces.\;$ Relative Topology $If\; (X,\tau)\; is\; topological\; space\; and\; A\subseteq X\; $ $Let\; \tau_{A}=\{A\cup U:U\in \tau\}\; is\;relative\;topology\;and\;(A,\tau_{A})\;is\;topological\;subspace.$](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjpGe9DnwI5WcbhBYHbolqcVSlQfDnhctLSDiMdBNdHjmurnXpcE3SLsVCJOT8kKn22l2PVN2X1UbujSGjcnCpyLfT1a1GyjpQ4QGWPtVi-HAkPGqRneL22179Fav8Jp8Belwnv2-1H5dx/s1600/topology-1.png "Topology Short Notes 1")
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