கச்சிதமான ஒழுங்கு வெளி
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 [[கணிதம்|கணிதத்தில்]], ஒரு இடவெளியில் உள்ள  வ்வொரு [[தொடர்வரிசை|முடிவற்ற  ஒழுங்கு வரிசை]] ஒரு ஒருங்கும் உள் ஒழுங்குவரிசையினை கொண்டிருப்பின் அது கச்சிதமான ஒழுங்கு இடவெளி எனப்படும்.
 மெட்ரிக் வெளியில் கச்சிதத் தன்மையும், கச்சிதமான ஒழுங்கு தன்மையும் ஒன்றுக்கொன்று சமமானதாக இருந்த பொழுதும், பொதுவான இடவெளியில் அவை இரண்டும் சமமானதாகக் கருத முடியாது. ஒரு மெட்ரிக் வெளி X -ல் உள்ள ஒவ்வொரு ஒழுங்குவரிசையும் மெட்ரிக் வெளி  X-ல் உள்ள ஒரு புள்ளியில் ஒருங்கும் உள் ஒழுங்குவரிசையினை கொண்டிருப்பின் அது கச்சிதமான(ஒழுங்கு) இடவெளி எனப்படும்.
 
 == சான்றாதாரம் ==
 * {{Cite book|last=[[James Munkres|Munkres, James]]|author=[[James Munkres|Munkres, James]]|year=1999|title=Topology|edition=2nd|publisher=[[Prentice Hall]]|isbn=0-13-181629-2|ISBN=0-13-181629-2}}
 * Steen, Lynn A. and Seebach, J. Arthur Jr.; ''Counterexamples in Topology'', Holt, Rinehart and Winston (1970). {{ISBN|0-03-079485-4}}0-03-079485-4.
 * {{Cite book|last=Willard, Stephen|author=Willard, Stephen|year=2004|title=General Topology|url=https://archive.org/details/generaltopology0000will|publisher=Dover Publications|isbn=0-486-43479-6|ISBN=0-486-43479-6}}
 &lt;div class=&quot;cx-template-editor-source-container&quot; dir=&quot;ltr&quot; style=&quot;display: none;&quot; lang=&quot;en&quot;&gt;&lt;div class=&quot;cx-template-editor-source&quot;&gt;&lt;div class=&quot;cx-template-editor-title&quot; title=&quot;This template formats a citation to a book using the provided bibliographic information (such as author and title) as well as various formatting options.&quot;&gt;Cite book&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;title&quot; class=&quot;cx-template-editor-param-key&quot;&gt;Title&lt;/span&gt;&lt;span data-key=&quot;title&quot; title=&quot;The title of the book; displays in italics&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;title&quot; style=&quot;position: relative;&quot;&gt;General Topology&lt;/div&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;last&quot; class=&quot;cx-template-editor-param-key&quot;&gt;Last name&lt;/span&gt;&lt;span data-key=&quot;last&quot; title=&quot;The surname of the author; don't wikilink, use 'authorlink'; can suffix with a numeral to add additional authors&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;last&quot; style=&quot;position: relative;&quot;&gt;Willard, Stephen&lt;/div&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;publisher&quot; class=&quot;cx-template-editor-param-key&quot;&gt;Publisher&lt;/span&gt;&lt;span data-key=&quot;publisher&quot; title=&quot;Name of the publisher; displays after title&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;publisher&quot; style=&quot;position: relative;&quot;&gt;Dover Publications&lt;/div&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;year&quot; class=&quot;cx-template-editor-param-key&quot;&gt;Year of publication&lt;/span&gt;&lt;span data-key=&quot;year&quot; title=&quot;Year of the source being referenced; use 'date' instead, if month and day are also known&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;year&quot; style=&quot;position: relative;&quot;&gt;2004&lt;/div&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;isbn&quot; class=&quot;cx-template-editor-param-key&quot;&gt;ISBN&lt;/span&gt;&lt;span data-key=&quot;isbn&quot; title=&quot;International Standard Book Number; use the 13-digit ISBN where possible&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;isbn&quot; style=&quot;position: relative;&quot;&gt;0-486-43479-6&lt;/div&gt;&lt;/div&gt;&lt;/div&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-source-container&quot; dir=&quot;ltr&quot; style=&quot;display: none;&quot; lang=&quot;en&quot;&gt;&lt;div class=&quot;cx-template-editor-source&quot;&gt;&lt;div class=&quot;cx-template-editor-title&quot; title=&quot;This template formats a citation to a book using the provided bibliographic information (such as author and title) as well as various formatting options.&quot;&gt;Cite book&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;title&quot; class=&quot;cx-template-editor-param-key&quot;&gt;Title&lt;/span&gt;&lt;span data-key=&quot;title&quot; title=&quot;The title of the book; displays in italics&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;title&quot; style=&quot;position: relative;&quot;&gt;General Topology&lt;/div&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;last&quot; class=&quot;cx-template-editor-param-key&quot;&gt;Last name&lt;/span&gt;&lt;span data-key=&quot;last&quot; title=&quot;The surname of the author; don't wikilink, use 'authorlink'; can suffix with a numeral to add additional authors&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;last&quot; style=&quot;position: relative;&quot;&gt;Willard, Stephen&lt;/div&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;publisher&quot; class=&quot;cx-template-editor-param-key&quot;&gt;Publisher&lt;/span&gt;&lt;span data-key=&quot;publisher&quot; title=&quot;Name of the publisher; displays after title&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;publisher&quot; style=&quot;position: relative;&quot;&gt;Dover Publications&lt;/div&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;year&quot; class=&quot;cx-template-editor-param-key&quot;&gt;Year of publication&lt;/span&gt;&lt;span data-key=&quot;year&quot; title=&quot;Year of the source being referenced; use 'date' instead, if month and day are also known&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;year&quot; style=&quot;position: relative;&quot;&gt;2004&lt;/div&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;isbn&quot; class=&quot;cx-template-editor-param-key&quot;&gt;ISBN&lt;/span&gt;&lt;span data-key=&quot;isbn&quot; title=&quot;International Standard Book Number; use the 13-digit ISBN where possible&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;isbn&quot; style=&quot;position: relative;&quot;&gt;0-486-43479-6&lt;/div&gt;&lt;/div&gt;&lt;/div&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-source-container&quot; dir=&quot;ltr&quot; style=&quot;display: none;&quot; lang=&quot;en&quot;&gt;&lt;div class=&quot;cx-template-editor-source&quot;&gt;&lt;div class=&quot;cx-template-editor-title&quot; title=&quot;This template formats a citation to a book using the provided bibliographic information (such as author and title) as well as various formatting options.&quot;&gt;Cite book&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;title&quot; class=&quot;cx-template-editor-param-key&quot;&gt;Title&lt;/span&gt;&lt;span data-key=&quot;title&quot; title=&quot;The title of the book; displays in italics&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;title&quot; style=&quot;position: relative;&quot;&gt;General Topology&lt;/div&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;last&quot; class=&quot;cx-template-editor-param-key&quot;&gt;Last name&lt;/span&gt;&lt;span data-key=&quot;last&quot; title=&quot;The surname of the author; don't wikilink, use 'authorlink'; can suffix with a numeral to add additional authors&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;last&quot; style=&quot;position: relative;&quot;&gt;Willard, Stephen&lt;/div&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;publisher&quot; class=&quot;cx-template-editor-param-key&quot;&gt;Publisher&lt;/span&gt;&lt;span data-key=&quot;publisher&quot; title=&quot;Name of the publisher; displays after title&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;publisher&quot; style=&quot;position: relative;&quot;&gt;Dover Publications&lt;/div&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;year&quot; class=&quot;cx-template-editor-param-key&quot;&gt;Year of publication&lt;/span&gt;&lt;span data-key=&quot;year&quot; title=&quot;Year of the source being referenced; use 'date' instead, if month and day are also known&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;year&quot; style=&quot;position: relative;&quot;&gt;2004&lt;/div&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;isbn&quot; class=&quot;cx-template-editor-param-key&quot;&gt;ISBN&lt;/span&gt;&lt;span data-key=&quot;isbn&quot; title=&quot;International Standard Book Number; use the 13-digit ISBN where possible&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;isbn&quot; style=&quot;position: relative;&quot;&gt;0-486-43479-6&lt;/div&gt;&lt;/div&gt;&lt;/div&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-source-container&quot; dir=&quot;ltr&quot; style=&quot;display: none;&quot; lang=&quot;en&quot;&gt;&lt;div class=&quot;cx-template-editor-source&quot;&gt;&lt;div class=&quot;cx-template-editor-title&quot; title=&quot;This template formats a citation to a book using the provided bibliographic information (such as author and title) as well as various formatting options.&quot;&gt;Cite book&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;title&quot; class=&quot;cx-template-editor-param-key&quot;&gt;Title&lt;/span&gt;&lt;span data-key=&quot;title&quot; title=&quot;The title of the book; displays in italics&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;title&quot; style=&quot;position: relative;&quot;&gt;General Topology&lt;/div&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;last&quot; class=&quot;cx-template-editor-param-key&quot;&gt;Last name&lt;/span&gt;&lt;span data-key=&quot;last&quot; title=&quot;The surname of the author; don't wikilink, use 'authorlink'; can suffix with a numeral to add additional authors&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;last&quot; style=&quot;position: relative;&quot;&gt;Willard, Stephen&lt;/div&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;publisher&quot; class=&quot;cx-template-editor-param-key&quot;&gt;Publisher&lt;/span&gt;&lt;span data-key=&quot;publisher&quot; title=&quot;Name of the publisher; displays after title&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;publisher&quot; style=&quot;position: relative;&quot;&gt;Dover Publications&lt;/div&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;year&quot; class=&quot;cx-template-editor-param-key&quot;&gt;Year of publication&lt;/span&gt;&lt;span data-key=&quot;year&quot; title=&quot;Year of the source being referenced; use 'date' instead, if month and day are also known&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;year&quot; style=&quot;position: relative;&quot;&gt;2004&lt;/div&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param&quot;&gt;&lt;div class=&quot;cx-template-editor-param-title&quot;&gt;&lt;span id=&quot;isbn&quot; class=&quot;cx-template-editor-param-key&quot;&gt;ISBN&lt;/span&gt;&lt;span data-key=&quot;isbn&quot; title=&quot;International Standard Book Number; use the 13-digit ISBN where possible&quot; class=&quot;cx-template-editor-param-desc&quot;&gt;&lt;/span&gt;&lt;/div&gt;&lt;div class=&quot;cx-template-editor-param-value&quot; data-key=&quot;isbn&quot; style=&quot;position: relative;&quot;&gt;0-486-43479-6&lt;/div&gt;&lt;/div&gt;&lt;/div&gt;&lt;/div&gt;