Topological Subspaces Review

Topological Subspaces Review

We will now review some of the recent material regarding subspaces of topological spaces.

  • Recall from the Initial Topologies page that if $X$ is a set, $\{ Y_i : i \in I \}$ is a collection of topological spaces, and $\{ f_i : i \in I \}$ is a collection of maps $f_i : X \to Y_i$ then the Initial Topology on $X$ induced by the maps $\{ f_i : i \in I \}$ is defined to be the coarsest topology on $X$ which makes each $f_i$ continuous.
  • To construct the initial topology on $X$ induced by a collection of maps $\{ f_i : i \in I \}$ we start with a map $f_i : X \to Y_i$. We looked at all of the open sets $U$ in $Y_i$. Their inverse images $f^{-1}(U)$ in $X$ will be declared open. We declare all such inverse images of open sets in $Y_i$ to be open in $X$. We repeat this process for all $i \in I$. A schematic is given below to produce the initial topology on $X$ induced by a single map $f$:
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  • We then looked at an important theorem which gave us a subbasis for the initial topology on $X$ induced by a collection of maps $\{ f_i : i \in I \}$ which was:
(1)
\begin{align} \quad \mathcal S = \{ f_i^{-1}(U) : U \in \tau_i \} \end{align}
  • On the Final Topologies page we saw that if $X$ is a set, $\{ Y_i : i \in I \}$ is a collection of topological spaces, and $\{ f_i : i \in I \}$ is a collection of maps $f_i : Y_i \to X$ then the Final Topology on $X$ induced by the maps $\{ f_i : i \in I \}$ is defined to be the finest topology on $X$ which makes each $f_i$ continuous.
  • To construct the final topology on $X$ induced by a collection of maps $\{ f_i : i \in I \}$ we start by looking at any subset $V$ of $X$. We then look at all of the inverse images $f_i^{-1}(V)$ for $i \in I$. If all of these inverse images $f_i^{-1}(V)$ are open, then we declare the set $V$ to be open in $X$. A schematic is given below for constructing the final topology on $X$ induced by an arbitrary collection of maps.
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  • We then looked at an important theorem which gave us an explicit form for the final topology on $X$ induced by a collection of maps $\{ f_i : i \in I \}$ which was:
(2)
\begin{align} \quad \tau = \{ U \subseteq X : f^{-1}(U) \in \tau_i, \: \forall i \in I \} \end{align}
  • On the Topological Subspaces page we looked at a new type of topology. If $(X, \tau)$ is a topological space and $A \subseteq X$ then we can define a Topological Subspace to be the space $(A, \tau_A)$ where the topology $\tau_A$ is called the Subspace Topology on $A$ (from $X$) and is defined as:
(3)
\begin{align} \quad \tau_A = \{ A \cap U : U \in \tau \} \end{align}
  • We verified that $\tau_A$ is indeed a topology on $A$.
(4)
\begin{align} \quad C = A \cap D \end{align}
  • On the Hereditary Properties of Topological Spaces page we said that a property for a topological space $(X, \tau)$ is Hereditary if every subspace $(A, \tau_A)$ also has that property. A property that is not hereditary is said to be Nonhereditary.
  • On the Heredity of First Countability on Topological Subspaces page we saw that first countability is hereditary. In proving this, we took an arbitrary point in $a \in A$ (which is also in $X$). We then looked at a countable local basis of $a$ in $X$. If we take the intersections of the local basis elements with $A$ then we showed that the resulting collecting is a countable local basis of $a$ in $A$.
  • On the Heredity of Second Countability on Topological Subspaces page we also saw that second countability is hereditary. In proving this, we took a countable basis of the topology on $X$. We took the intersections of the basis elements with $A$ to produce a countable basis for the subspace topology on $A$.
  • On the Nonheredity of Separability on Topological Subspaces page we saw that while many properties are hereditary, some, such as separability, are not nonhereditary. We took our main topological space to be the product of two lower limit topological spaces, and gave the line $L : \{ (-x, x) : x \in \mathbb{R} \}$ the subspace topology. We then saw the line $L$ with the subspace topology was actually $L$ with the discrete topology. Therefore every singleton set, i.e., every point on the line $L$ is open which means that every dense subset of $L$ must contain all $L$. However, there are an uncountably infinite number of points on $L$ which means that $L$ has no countable and dense subset, i.e., $L$ is not separable.
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