FoldUnfold Table of Contents Open and Closed Maps on Topological Spaces Open and Closed Maps on Topological Spaces We have looked quite a bit at continuous maps on topological spaces, so we will turn out attention to looking at other types of maps on topological spaces - in particular, open maps and closed maps which we define below. Definition: Let $X$ and $Y$ be … [Read more...]

## Final Topologies

FoldUnfold Table of Contents Final Topologies Final Topologies Recall from the Initial Topologies page that if $X$ is a set, $Y$ is a topological space, and $f : X \to Y$ then the initial topology induced by $f$ on $X$ is the coarsest topology which makes the map $f : X \to Y$ continuous. More generally, if $X$ is a set, $\{ Y_i : i \in I \}$ is a collection of … [Read more...]

## Initial Topologies

FoldUnfold Table of Contents Initial Topologies Initial Topologies Now that we have looked at the definition of continuity of maps on topological spaces, we are now ready to look at some new topologies that are based off of the concept. We begin by describing a special type of topology on a set $X$ that is induced by a topological space $Y$ by a function $f : X \to Y$ (or … [Read more...]

## Summary of Equivalent Statements Regarding Continuous Maps on Topological Spaces

FoldUnfold Table of Contents Summary of Equivalent Statements Regarding Continuous Maps on Topological Spaces Summary of Equivalent Statements Regarding Continuous Maps on Topological Spaces We have see many different definitions/equivalent statements for a map $f : X \to Y$ (where $X$ and $Y$ are topological spaces) to be continuous on all of $X$. Be sure to review the … [Read more...]

## The Closed Set Definition of Continuous Maps on Topological Spaces

FoldUnfold Table of Contents The Closed Set Definition of Continuous Maps on Topological Spaces The Closed Set Definition of Continuous Maps on Topological Spaces Recall from the Continuous Maps on Topological Spaces page that we say $f : X \to Y$ is continuous at $a \in X$ if there exists local bases $\mathcal B_a$ of $a$ and $\mathcal B_{f(a)}$ of $f(a)$ such that for … [Read more...]

## Equivalent Statements Regarding Continuous Maps on Topological Spaces

FoldUnfold Table of Contents Equivalent Statements Regarding Continuous Maps on Topological Spaces Equivalent Statements Regarding Continuous Maps on Topological Spaces Recall from the Continuous Maps on Topological Spaces page that if $X$ and $Y$ are topological spaces then a map $f : X \to Y$ is said to be continuous at the point $a \in X$ if there exists local bases … [Read more...]

## The Open Neighbourhood Definition of Continuous Maps on Topological Spaces

FoldUnfold Table of Contents The Open Neighbourhood Definition of Continuous Maps on Topological Spaces The Open Neighbourhood Definition of Continuous Maps on Topological Spaces Recall from the Continuous Maps on Topological Spaces page that if $X$ and $Y$ are topological spaces then a function $f : X \to Y$ is said to be Continuous at a point $a \in X$ if there exists … [Read more...]

## Continuity of the Composition of Continuous Maps on Topo. Spaces

FoldUnfold Table of Contents Continuity of the Composition of Continuous Maps on Topo. Spaces Continuity of the Composition of Continuous Maps on Topo. Spaces On the Summary of Equivalent Statements Regarding Continuous Maps on Topological Spaces page we summarizes all of the equivalent states for a map $f : X \to Y$ on topological spaces to be continuous on all of $X$. … [Read more...]

## Continuous Mappings on Topological Spaces

FoldUnfold Table of Contents Continuous Mappings on Topological Spaces Continuous Mappings on Topological Spaces Recall from the Local Bases of a Point in a Topological Space page that if $(X, \tau)$ is a topological space then a local basis of a point $x \in X$ is a collection $\mathcal B_x$ of open neighbourhoods of $x$ such that for each $U \in \tau$ with $x \in U$ … [Read more...]

## Generalizing Continuity to Maps on Topological Spaces

FoldUnfold Table of Contents Generalizing Continuity to Maps on Topological Spaces Generalizing Continuity to Maps on Topological Spaces The reader should be somewhat familiar with the definition of continuity of a function $f$ mapping $\mathbb{R}$ into itself. Recall that $f$ is said to be continuous at $a \in \mathbb{R}$ if for all $\epsilon > 0$ there exists a $\delta … [Read more...]