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What all topological properties are preserved under attaching a 2-cell?

Relations between various definitions of a Radon measureUnder what conditions the quotient space of a manifold is a manifold?Hausdorffness of quotient spaceShow the cone on the integers is not locally compactFind a (simple?) counterexample to this statement about topological manifolds.Is $mathbbR/mathordsim$ a Hausdorff space if $(x,y)!:xsim y$ is a closed subset of $mathbbRtimesmathbbR$?Does collapsing the connected components of a topological space make it totally disconnected?Show the restriction of a certain quotient map is closedQuotient space and quotient set for $mathrmEnd(mathbbR^2)$On the quotient space $X^n/S_n$, for a Hausdorff, contractible, locally path connected topological space $X$

0

$begingroup$

Let $Y$ be a topological space.

Let $f:mathbbS^1=partialmathbbD^2rightarrow Y$ be a continuous map.

By attaching $2$-cell to $Y$ we mean the space $Ybigsqcup mathbbD^2$ under the identification that $xin mathbbS^1$ is identified with its image $f(x)$ in $Y$.

Let us denote the quotient space $(Ybigsqcup mathbbD^2)/sim$ by $X$.

Question : What all properties are preserved under attaching a $2$-cell?

I am interested (but not limited to) Hausdorff, locally Hausdorff, regular, normal, compact, paracompact, contractible, locally contractible.

general-topology algebraic-topology quotient-spaces

share|cite|improve this question

edited Mar 22 at 16:32

Praphulla Koushik

asked Mar 22 at 7:12

Praphulla KoushikPraphulla Koushik

203119

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It is not necessary that you say all properties preserved under attaching. I would be thankful even if you say one in your answer... :)
  $endgroup$
  – Praphulla Koushik
  Mar 22 at 7:13
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you mean $S^0$ instead of $S^1$, and $Ybigsqcup D^1$ instead of $Ybigsqcup partial D^1$?
  $endgroup$
  – Eric Wofsey
  Mar 22 at 8:02
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I have problem with names.. Does it look ok now? @EricWofsey
  $endgroup$
  – Praphulla Koushik
  Mar 22 at 8:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Not all topological properties are preserved by attaching a 2-cell. For example, if $Y$ is a single point, then you get the 2-sphere, which is not contractible, does not have the same homotopy groups of a point, or homology groups, or cohomology groups. So, be careful ...
  $endgroup$
  – Laz
  Mar 22 at 23:30
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  For example, compactness is trivial since your adjunction space is a quotient of a disjoint union of $Y$ compact and the 2-sphere compact. For Hausdoffness, and regularness, check out Hatcher's book, in the appendix about CW complexes.
  $endgroup$
  – Laz
  Mar 22 at 23:49
  
  

  
  
  
  

 | 
show 1 more comment

0

$begingroup$

Let $Y$ be a topological space.

Let $f:mathbbS^1=partialmathbbD^2rightarrow Y$ be a continuous map.

By attaching $2$-cell to $Y$ we mean the space $Ybigsqcup mathbbD^2$ under the identification that $xin mathbbS^1$ is identified with its image $f(x)$ in $Y$.

Let us denote the quotient space $(Ybigsqcup mathbbD^2)/sim$ by $X$.

Question : What all properties are preserved under attaching a $2$-cell?

I am interested (but not limited to) Hausdorff, locally Hausdorff, regular, normal, compact, paracompact, contractible, locally contractible.

general-topology algebraic-topology quotient-spaces

share|cite|improve this question

edited Mar 22 at 16:32

Praphulla Koushik

asked Mar 22 at 7:12

Praphulla KoushikPraphulla Koushik

203119

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It is not necessary that you say all properties preserved under attaching. I would be thankful even if you say one in your answer... :)
  $endgroup$
  – Praphulla Koushik
  Mar 22 at 7:13
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you mean $S^0$ instead of $S^1$, and $Ybigsqcup D^1$ instead of $Ybigsqcup partial D^1$?
  $endgroup$
  – Eric Wofsey
  Mar 22 at 8:02
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I have problem with names.. Does it look ok now? @EricWofsey
  $endgroup$
  – Praphulla Koushik
  Mar 22 at 8:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Not all topological properties are preserved by attaching a 2-cell. For example, if $Y$ is a single point, then you get the 2-sphere, which is not contractible, does not have the same homotopy groups of a point, or homology groups, or cohomology groups. So, be careful ...
  $endgroup$
  – Laz
  Mar 22 at 23:30
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  For example, compactness is trivial since your adjunction space is a quotient of a disjoint union of $Y$ compact and the 2-sphere compact. For Hausdoffness, and regularness, check out Hatcher's book, in the appendix about CW complexes.
  $endgroup$
  – Laz
  Mar 22 at 23:49
  
  

  
  
  
  

 | 
show 1 more comment

$begingroup$

Let $Y$ be a topological space.

Let $f:mathbbS^1=partialmathbbD^2rightarrow Y$ be a continuous map.

By attaching $2$-cell to $Y$ we mean the space $Ybigsqcup mathbbD^2$ under the identification that $xin mathbbS^1$ is identified with its image $f(x)$ in $Y$.

Let us denote the quotient space $(Ybigsqcup mathbbD^2)/sim$ by $X$.

Question : What all properties are preserved under attaching a $2$-cell?

I am interested (but not limited to) Hausdorff, locally Hausdorff, regular, normal, compact, paracompact, contractible, locally contractible.

general-topology algebraic-topology quotient-spaces

share|cite|improve this question

edited Mar 22 at 16:32

Praphulla Koushik

asked Mar 22 at 7:12

Praphulla KoushikPraphulla Koushik

203119

$endgroup$

share|cite|improve this question

edited Mar 22 at 16:32

Praphulla Koushik

asked Mar 22 at 7:12

Praphulla KoushikPraphulla Koushik

203119

share|cite|improve this question

edited Mar 22 at 16:32

Praphulla Koushik

asked Mar 22 at 7:12

Praphulla KoushikPraphulla Koushik

203119

asked Mar 22 at 7:12

Praphulla KoushikPraphulla Koushik

203119

asked Mar 22 at 7:12

Praphulla KoushikPraphulla Koushik

203119