https://topospaces.subwiki.org/w/index.php?action=history&feed=atom&title=Monotonically_normal_implies_hereditarily_normal 2026-07-20T07:45:54Z Revision history for this page on the wiki MediaWiki 1.41.2 https://topospaces.subwiki.org/w/index.php?title=Monotonically_normal_implies_hereditarily_normal&diff=2879&oldid=prev 2009-10-24T23:09:54Z

<p>Created page with &#039;{{topospace property implication| stronger = monotonically normal space| weaker = hereditarily normal space}} ==Statement== Any <a href="/wiki/Monotonically_normal_space" title="Monotonically normal space">monotonically normal space</a> is a [[hereditari…&#039;</p> <p><b>New page</b></p><div>{{topospace property implication|<br /> stronger = monotonically normal space|<br /> weaker = hereditarily normal space}}<br /> <br /> ==Statement==<br /> <br /> Any [[monotonically normal space]] is a [[hereditarily normal space]]: every subspace of the space is a [[normal space]].<br /> <br /> ==Related facts==<br /> <br /> * [[Monotone normality is hereditary]]<br /> * [[Monotonically normal implies collectionwise normal]]<br /> * [[Monotonically normal implies hereditarily collectionwise normal]]<br /> <br /> ==Facts used==<br /> <br /> # [[uses::Monotone normality is hereditary]]<br /> # [[uses::Monotonically normal implies normal]]<br /> <br /> ==Proof==<br /> <br /> &#039;&#039;&#039;Given&#039;&#039;&#039;: A monotonically normal space &lt;math&gt;X&lt;/math&gt;.<br /> <br /> &#039;&#039;&#039;To prove&#039;&#039;&#039;: Every subspace of &lt;math&gt;X&lt;/math&gt; is normal.<br /> <br /> &#039;&#039;&#039;Proof&#039;&#039;&#039;: By fact (1), every subspace of &lt;math&gt;X&lt;/math&gt; is monotonically normal. Using fact (2), we obtain that every subspace of &lt;math&gt;X&lt;/math&gt; is normal.</div>

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