Why is $X$ connected if $A$ and $X/A$ are connected in a topological group?

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  • Thread starter Euge
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    2017
In summary, a topological group is a combination of a group and a topological space where the group operation is continuous with respect to the topology. The connectedness of $X$ is important because it ensures that the group operation is continuous, and the connectedness of $A$ and $X/A$ means that they cannot be split into disjoint open sets. The connectedness of $X$ implies the connectedness of $A$ and $X/A$ but the converse is not always true.
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Euge
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Here is this week's POTW:

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Let $X$ be a topological group; let $A$ be a subgroup of $X$ such that $A$ and $X/A$ are connected. Show that $X$ is connected.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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No one answered this week's problem. You can read my solution below.
Suppose $\{U,V\}$ is a separation of $X$. The natural projection map $p: X \to X/A$ is open, so since $p(U) \cup p(V) = X/A$ and $X/A$ is connected, then $p(U) \cap p(V)$ is nonempty. That means there exists $x\in X$, $u\in U$, and $v\in V$ such that $u,v\in xA$. Then $\{U\cap xA, V\cap xA\}$ separates $xA$, which is impossible since $xA$ is connected (as $A$ is connected and left translation by $x$ is a homeomorphism of $X$).
 

FAQ: Why is $X$ connected if $A$ and $X/A$ are connected in a topological group?

What is a topological group?

A topological group is a mathematical structure that combines the properties of a group (a set with a binary operation) and a topological space (a set with a notion of "closeness"). In a topological group, the group operation is continuous with respect to the topology, meaning that small changes in the inputs result in small changes in the output.

Why is the connectedness of $X$ important in this context?

The connectedness of $X$ is important because it tells us about the structure of the topological group. In particular, it tells us that $X$ cannot be split into two disjoint open sets, which is a property that is necessary for the group operation to be continuous. Without this property, the group would not be a topological group.

What does it mean for $A$ and $X/A$ to be connected?

When we say that $A$ and $X/A$ are connected, we mean that they are both connected topological spaces. This means that they cannot be split into two disjoint open sets, and that any continuous map from them to a discrete set (a set with the indiscrete topology) is constant. In other words, they are "uncuttable" and have no "holes" in them.

Why does the connectedness of $A$ and $X/A$ imply the connectedness of $X$?

By definition, the quotient space $X/A$ is obtained by identifying all points in $A$ to a single point. This "collapsing" of $A$ does not change the connectedness of $X$, as any open set in $X$ will still be open after identifying the points in $A$. Similarly, $A$ itself cannot be cut into two disjoint open sets, and thus the connectedness of $X$ is preserved.

Is the converse true, i.e. does the connectedness of $X$ imply the connectedness of $A$ and $X/A$?

No, the converse is not necessarily true. Just because $X$ is connected does not mean that $A$ and $X/A$ are connected. For example, consider the group $X = \mathbb{R}$ with the subgroup $A = \mathbb{Z}$. While $X$ is connected, $A$ is not, as it can be split into the two disjoint open sets $(\infty, 0)$ and $(0, \infty)$. Similarly, $X/A$ is not connected, as it consists of two points (the equivalence classes of $0$ and $1$). However, if we restrict our attention to $X \setminus A$, then the connectedness of $X$ does imply the connectedness of $A$ and $X/A$.

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