Pointed space

<h2 id="category-of-pointed-spaces">Category of pointed spaces</h2>
<p>The <a href="/facts/Class_(set_theory)/F5EmDd9b">class</a> of all pointed spaces forms a <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> Top
  
    
      
        ∙
      
    
    {\displaystyle \bullet }
  
 with basepoint preserving continuous maps as <a href="/facts/Morphism/Kdpuyz3S">morphisms</a>. Another way to think about this category is as the <a href="/facts/Comma_category/wTa3tURp">comma category</a>, (
  
    
      
        {
        ∙
        }
        ↓
      
    
    {\displaystyle \{\bullet \}\downarrow }
  
 Top) where 
  
    
      
        {
        ∙
        }
      
    
    {\displaystyle \{\bullet \}}
  
 is any one point space and Top is the <a href="/facts/Category_of_topological_spaces/qH4SaEWa">category of topological spaces</a>. (This is also called a <a href="/facts/Coslice_category/wTa3tURp">coslice category</a> denoted 
  
    
      
        {
        ∙
        }
        
          /
        
      
    
    {\displaystyle \{\bullet \}/}
  
Top.) Objects in this category are continuous maps 
  
    
      
        {
        ∙
        }
        →
        X
        .
      
    
    {\displaystyle \{\bullet \}\to X.}
  
 Such maps can be thought of as picking out a basepoint in 
  
    
      
        X
        .
      
    
    {\displaystyle X.}
  
 Morphisms in (
  
    
      
        {
        ∙
        }
        ↓
      
    
    {\displaystyle \{\bullet \}\downarrow }
  
 Top) are morphisms in Top for which the following diagram <a href="/facts/Commutative_diagram/bdG8xuer">commutes</a>:
</p>

<p>It is easy to see that commutativity of the diagram is equivalent to the condition that 
  
    
      
        f
      
    
    {\displaystyle f}
  
 preserves basepoints.
</p><p>As a pointed space, 
  
    
      
        {
        ∙
        }
      
    
    {\displaystyle \{\bullet \}}
  
 is a <a href="/facts/Zero_object/oeckvXRj">zero object</a> in Top
  
    
      
        {
        ∙
        }
      
    
    {\displaystyle \{\bullet \}}
  
, while it is only a <a href="/facts/Terminal_object/oeckvXRj">terminal object</a> in Top.
</p><p>There is a <a href="/facts/Forgetful_functor/onEwu6HA">forgetful functor</a> Top
  
    
      
        {
        ∙
        }
      
    
    {\displaystyle \{\bullet \}}
  
 
  
    
      
        →
      
    
    {\displaystyle \to }
  
 Top which "forgets" which point is the basepoint. This functor has a <a href="/facts/Adjoint_functor/bJ1P7AN2">left adjoint</a> which assigns to each topological space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 the <a href="/facts/Disjoint_union/x9QSfkr5">disjoint union</a> of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and a one-point space 
  
    
      
        {
        ∙
        }
      
    
    {\displaystyle \{\bullet \}}
  
 whose single element is taken to be the basepoint.
</p>
<h2 id="operations-on-pointed-spaces">Operations on pointed spaces</h2>
<ul><li>A subspace of a pointed space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a <a href="/facts/Subspace_(topology)/NjU6z4Ca">topological subspace</a> 
  
    
      
        A
        ⊆
        X
      
    
    {\displaystyle A\subseteq X}
  
 which shares its basepoint with 
  
    
      
        X
      
    
    {\displaystyle X}
  
 so that the <a href="/facts/Inclusion_map/VUFNdaDH">inclusion map</a> is basepoint preserving.</li>
<li>One can form the <a href="/facts/Quotient_space_(topology)/muE0qQVC">quotient</a> of a pointed space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 under any <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a>. The basepoint of the quotient is the image of the basepoint in 
  
    
      
        X
      
    
    {\displaystyle X}
  
 under the quotient map.</li>
<li>One can form the <a href="/facts/Product_(category_theory)/evFecz0t">product</a> of two pointed spaces 
  
    
      
        
          (
          
            X
            ,
            
              x
              
                0
              
            
          
          )
        
        ,
      
    
    {\displaystyle \left(X,x_{0}\right),}
  
 
  
    
      
        
          (
          
            Y
            ,
            
              y
              
                0
              
            
          
          )
        
      
    
    {\displaystyle \left(Y,y_{0}\right)}
  
 as the <a href="/facts/Product_(topology)/0hg02IaX">topological product</a> 
  
    
      
        X
        ×
        Y
      
    
    {\displaystyle X\times Y}
  
 with 
  
    
      
        
          (
          
            
              x
              
                0
              
            
            ,
            
              y
              
                0
              
            
          
          )
        
      
    
    {\displaystyle \left(x_{0},y_{0}\right)}
  
serving as the basepoint.</li>
<li>The <a href="/facts/Coproduct/0bJ8SCdu">coproduct</a> in the category of pointed spaces is the <a href="/facts/Wedge_sum/mhBoNDAx">wedge sum</a>, which can be thought of as the 'one-point union' of spaces.</li>
<li>The <a href="/facts/Smash_product/3gLz9z6f">smash product</a> of two pointed spaces is essentially the <a href="/facts/Quotient_space_(topology)/muE0qQVC">quotient</a> of the direct product and the wedge sum. We would like to say that the smash product turns the category of pointed spaces into a <a href="/facts/Symmetric_monoidal_category/AxhumNny">symmetric monoidal category</a> with the pointed <a href="/facts/0-sphere/bFu2S7E2">0-sphere</a> as the unit object, but this is false for general spaces: the associativity condition might fail. But it is true for some more restricted categories of spaces, such as <a href="/facts/Compactly_generated_space/2A5umYGa">compactly generated</a> <a href="/facts/Weak_Hausdorff_space/2zl8Igzv">weak Hausdorff</a> ones.</li>
<li>The <a href="/facts/Reduced_suspension/pIWr58dM">reduced suspension</a> 
  
    
      
        Σ
        X
      
    
    {\displaystyle \Sigma X}
  
 of a pointed space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is (up to a <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphism</a>) the smash product of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and the pointed circle 
  
    
      
        
          S
          
            1
          
        
        .
      
    
    {\displaystyle S^{1}.}
  
</li>
<li>The reduced suspension is a functor from the category of pointed spaces to itself. This functor is <a href="/facts/Left_adjoint/bJ1P7AN2">left adjoint</a> to the functor 
  
    
      
        Ω
      
    
    {\displaystyle \Omega }
  
 taking a pointed space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 to its <a href="/facts/Loop_space/S5xkdCkx">loop space</a> 
  
    
      
        Ω
        X
      
    
    {\displaystyle \Omega X}
  
.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Category_of_groups/sxSd9Kbc">Category of groups</a> – category of groups and group homomorphismsPages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Category_of_metric_spaces/W2kbarEJ">Category of metric spaces</a> – mathematical category with metric spaces as its objects and distance-non-increasing maps as its morphismsPages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Category_of_sets/LF5ykYl0">Category of sets</a> – Category in mathematics where the objects are sets</li>
<li><a href="/facts/Category_of_topological_spaces/qH4SaEWa">Category of topological spaces</a> – category whose objects are topological spaces and whose morphisms are continuous mapsPages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Category_of_topological_vector_spaces/NlyMIzQM">Category of topological vector spaces</a> – Topological category</li></ul>

<ul><li>Gamelin, Theodore W.; Greene, Robert Everist (1999) [1983]. <a href="https://archive.org/details/introductiontoto00game"><i>Introduction to Topology</i></a> (second ed.). <a href="/facts/Dover_Publications/RwjvlSFv">Dover Publications</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-40680-6.</li>
<li><a href="/facts/Saunders_Mac_Lane/rjgUqfkA">Mac Lane, Saunders</a> (September 1998). <i><a href="/facts/Categories_for_the_Working_Mathematician/SLQDat9B">Categories for the Working Mathematician</a></i> (second ed.). Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-98403-8.</li></ul>
<ul><li><a href="https://mathoverflow.net/q/40945">mathoverflow discussion on several base points and groupoids </a></li></ul>

Pointed space open-in-new

Pointed space